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Sunday, July 27, 2014

Field Day #3

Our Field Day's this summer have been such a success that this past week the kids took part in their third Field Day! We brought back two favorites (bag races and slip n' slide races), tried a few new races, and played a game. A great morning was had by all and snoring was a big part of the afternoon.

The kids were super excited to shoot with the bow and arrow (not a real arrow for those concerned). When I told them the day before about shooting with the bow and arrow, their response was "finally! We've been waiting all summer for that!" The bow and arrow proved very difficult for them (Max was the only one who could figure it out), but they still had a great time trying.

The kids saw a bunny in the yard, stopped their race, and chased the bunny around the backyard. Max got the bow and arrow all ready "to catch that bunny!"

Matt showed the kids how to use the ribbon dancer.....it was extra special. The kids had a great time taking turns (each turn was 5 minutes long) and dancing around with the ribbon.

I got this from Target a few weeks ago for 75% off. This was an easy game for the kids to figure out, although I did have to go over safety rules of the game several times so no one got hurt with the golf balls.

The kids LOVE bag races. These took up at least a half hour of our morning. Elizabeth got to do bag races for the first time and got some help from mom (that'd be me).

She was all smiles and giggles as she raced.

Lawn mower races. Once again I used a jump rope to indicate the finish line.

The first ever cozy coupe race! The kids took turn pushing each other in the cozy coupes while running their fastest. I had to stay at the finish line and help stop the cars before the cars (and the kids in them) went into the swing set. The kids informed me this was one of their favorite races.

It may not be a race, but the kids colored the driveway with water sidewalk chalk. I filled a cup with water and the kids dipped their sidewalk chalk into the water. They liked how bright the chalk was. This kept the kids busy while they waited their turn for the slip n' slide races.

The kids had duel races and shaving cream slide races. We did the slip n' slide in the front yard this time around since the backyard is still recovering from the first time the kids did the slip n' slide. I required the hubs help with this activity since we did it in the front yard, but the kids did a great job of not running out into the street (my biggest fear) and following directions.

Elizabeth was not thrilled with the shaving cream slide!

The kids had such a busy morning running around and sliding in the water that they crashed for naps. I giggled to myself when I heard them snoring over the monitor and went in to find them crashed out on their cots with drool hanging out of their mouths! I think that's a pretty good indication of a successful morning!


Eidos-Montreal has announced Deus Ex Universe:

Loading

“ An ongoing, expanding and connected game world built across a generation of core games.

We first heard about Deus Ex Universe through a trademark filing spotted last month, but up until now we didn't know anything about it. In a blog post , studio head David Anfossi described Deus Ex Universe as "an ongoing, expanding and connected game world built across a generation of core games."It continues, "It’s a commitment on our part to deliver meaningful content that expands the franchise on a regular basis and to deliver a deep conspiracy that will span several connected Deus Ex games, creating a more immersive and richer experience than ever before.Eidos-Montreal has apparently started production on the first product in Deus Ex Universe; a new game for PC and next-generation consoles. Anfossi also added that most of the team who worked on Deus Ex: Human Revolution have already jumped over to the new title.While we may not have a name for it just yet, we have been given a piece of concept art that supposedly shows "shows trans-humanism segregation, which is a backdrop to our vision for the next Deus Ex. It represents a "ghetto-city' voluntarily built in order to separate the classes. The people in this segregated class have reshaped their environment, nostalgic for their ideal of Cyber Renaissance. This dark and dystopian vision sets the tone for things to come in Deus Ex."Given all the talk of "an ongoing, expanding and connected game world" that'll expand "on a regular basis", the temptation is to assume this may mean a Deus Ex MMO could be on the way in the near future. Until Eidos-Montreal gives us more details though, we're just going to have to be patient.To tide us over however, the studio has announced the long-awaited Deus Ex: Human Revolution Director's Cut will be available on Xbox 360, PlayStation 3, Wii U and PC on October 22 in North America and on October 25 in Europe and Australia.

Luke Karmali is IGN's UK Junior Editor and big MMO fan. You too can revel in mediocrity by following him on IGN and on Twitter


Accelerated detection with prospective surveillance for cutaneous malignant melanoma in high-risk groups.
In 1983 a classification scheme was proposed for patients with atypical naevi, according to their personal and family history of melanoma and atypical naevi. To assess the predictive value of these features we undertook prospective surveillance of patients at high risk of primary melanoma. We followed up 116 patients each with 3 or more clinically atypical naevi for at least 5 years. Patients are examined and naevi are photographed every 3-6 months; lesions showing disturbing change are excised for histopathology. Among 85 patients with no personal or family history of melanoma, 5 invasive (level 2 or deeper) melanomas developed during 583 person-years of follow up. The expected number of invasive melanomas in this population would be 0.054; the increased risk is significant (p < 0.001; relative risk 92 [95% CI 30-216]). There was a similarly increased risk of new melanoma also among 24 patients with atypical naevi plus a history of previous melanoma (observed 2, expected 0.022, p < 0.001; relative risk 91 [11-328]). By comparison, no second melanoma developed among 25 patients with previous melanoma but a normal naevus pattern during 213 person-years of similarly intensive follow-up. The risk of melanoma was highest among 7 patients with atypical naevi and a family history of melanoma (observed 6, expected 0.009, p < 0.001; relative risk 444 [121-1138]). The median thickness of surveillance-detected melanomas was 0.75 mm (range 0.40-1.05 mm) in this group. This study shows the value of clinical follow-up of high-risk patients to detect early thin melanomas.


Q:

How to remove duplicate nested objects Django REST?

How to remove duplicate nested comments Django REST?
Serializer recoursive displays comments.
In JSON duplicates comments (the third is displayed in two places):
How to get rid of it?
{
    "id": 1,
    "slug": "mp3kit",
    "publications_date": "2019-03-19T13:49:58Z",
    "tag": "Tag",
    "title": "Title",
    "content_preview": "Content preview",
    "body": "Body",
    "image_preview": null,
    "image_preview_name": "mp3kit",
    "comments": [
        {
            "name": "First",
            "body": "First",
            "date": "2019-06-02T15:09:16.788405Z",
            "id": "57fd0069-a8ce-484f-afac-4cc609a1b70c",
            "parent": null,
            "reply": [
                {
                    "name": "Third",
                    "body": "Third",
                    "date": "2019-06-02T15:09:42.512872Z",
                    "id": "0801ef96-796d-477b-8631-b21559527156",
                    "parent": "57fd0069-a8ce-484f-afac-4cc609a1b70c",
                    "reply": []
                }
            ]
        },
        {
            "name": "Second",
            "body": "Second",
            "date": "2019-06-02T15:09:27.209497Z",
            "id": "5b2f0d25-bcd0-4b19-b05c-e6a5e7de1280",
            "parent": null,
            "reply": []
        },
        {
            "name": "Third",
            "body": "Third",
            "date": "2019-06-02T15:09:42.512872Z",
            "id": "0801ef96-796d-477b-8631-b21559527156",
            "parent": "57fd0069-a8ce-484f-afac-4cc609a1b70c",
            "reply": []
        }
    ]
}

As I understand you need to put a filter on the field "comment" in ArticleSerializer, to disable output comments with parent?
Model:
class Article(models.Model):
    id = models.AutoField(primary_key=True)
    slug = models.SlugField()
    publications_date = models.DateTimeField(blank=True, null=True, default=datetime.datetime.now)
    tag = models.CharField(max_length=100, blank=True, null=True)
    title = models.CharField(max_length=100, blank=True, null=True)
    content_preview = models.TextField(blank=True)
    body = HTMLField(blank=True, null=True)
    image_preview = models.ImageField(upload_to=upload_to, blank=True)
    image_preview_name = models.CharField(max_length=100, blank=True, null=True)

    def __str__(self):
        return self.title

    @property
    def comments_list(self):
        return self.comments.filter()

class Comment(models.Model):
    id = models.CharField(max_length=100, blank=True, unique=True, default=uuid.uuid4, primary_key=True)
    date = models.DateTimeField(auto_now=True, auto_now_add=False)
    name = models.CharField(max_length=100)
    body = models.TextField(blank=True, null=True)
    post = models.ForeignKey(Article, on_delete=models.CASCADE, blank=True,
                             related_name='comments', related_query_name='comment')
    parent = models.ForeignKey('self', related_name='reply', null=True, blank=True, on_delete=models.PROTECT)

    def __str__(self):
        return f'Post - "{self.post.title}", Body - "{self.body}"'

Serializer:
class RecursiveSerializer(serializers.Serializer):
    def to_representation(self, value):
        serializer = self.parent.parent.__class__(value, context=self.context)
        return serializer.data

class CommentSerializer(serializers.ModelSerializer):
    """DRF Serializer For Listing Published Comment"""
    reply = RecursiveSerializer(many=True, read_only=True)

    class Meta:
        model = Comment
        fields = [
            'name',
            'body',
            'date',
            'id',
            'parent',
            'reply'
        ]

class ArticleSerializer(serializers.ModelSerializer):
    comments = CommentSerializer(many=True)

    class Meta:
        model = Article
        lookup_field = 'slug'
        fields = (
            'id',
            'slug',
            'publications_date',
            'tag',
            'title',
            'content_preview',
            'body',
            'image_preview',
            'image_preview_name',
            'comments'
        )

Views:
class DetailArticle(generics.RetrieveUpdateDestroyAPIView):
    queryset = Article.objects.all()
    serializer_class = ArticleSerializer
    lookup_field = 'slug'

A:

You can achieve that by using a SerializerMethodField inside your ArticleSerializer, like this:
class ArticleSerializer(serializers.ModelSerializer):
    comments = serializers.SerializerMethodField()

    def get_comments(self, instance):
        return CommentSerializer(instance.comments.filter(parent__isnull=True), many=True).data

    class Meta:
        model = Article
        lookup_field = 'slug'
        fields = (
            'id',
            'slug',
            'publications_date',
            'tag',
            'title',
            'content_preview',
            'body',
            'image_preview',
            'image_preview_name',
            'comments'
        )


Skovfarmen’s Brown Nooroo NoName

About

Glenn now lives with Sofia in Sweden. There are big plans (obedience, tracking, nose work and agility) for this little guy but the most important one is to be Sofias new best mate and companion.
You can follow the adventures of Sofia and Glenn at Sofias Blog


List of Mexican films of 1943

A list of the films produced in Mexico in 1943 (see 1943 in film):

1943

External links

1943
Films
Category:Lists of 1943 films by country or language


Alleged Puppy Doe abuser arraigned on new charges

The man accused of torturing Puppy Doe has pleaded innocent to charges he stole from an elderly Quincy woman in his care.

Patrick Ronan

The man accused of torturing Puppy Doe has pleaded innocent to charges he stole from an elderly Quincy woman in his care.

Radoslaw Czerkawski, 32, a Polish national, was arraigned Tuesday in Dedham Superior Court on two counts of larceny of property valued at over $250, one count of larceny of property valued at over $250 from a person over the age of 60 and one count of attempted larceny. Prosecutors allege that Czerkawski stole from the late Janina Stock, 95, of 89 Whitwell St. after he was hired by Stock's family to be her in-home caretaker.

The Norfolk County District Attorney's office said Czerkawski pretended Stock, who suffered from dementia, was his grandmother in order to convert $130,000 of her savings bonds into cash he could use.

Czerkawsi was indicted and arraigned in Norfolk Superior Court in December on a dozen counts of animal cruelty, in a case that drew worldwide attention and inspired numerous vigils by pet owners and animal rights advocates.

On Tuesday, dozens of people gathered outside Dedham Superior Court to rally for tougher penalties against animal abusers, as they have at all of Czerkawski's court appearances since his arrest in October.

Police said Stock died of natural causes on Aug. 31, the same day a brutally abused Puppy Doe was found on Carroll's Lane in Quincy. Quincy police and the Animal Rescue League asked for the public's help in finding the person responsible for torturing and starving the young female pit bull so badly that the dog had to be euthanized.

Czerkawski – who bought the dog from on Craigslist – was arrested in Connecticut in late October, after police matched him to DNA found in the Whitwell Street house where he lived with the elderly woman – who died the same day that Puppy Doe was found.

Czerkawski also faces separate charges out of New Bedford after he was accused of stealing about $6,000 from a church he was staying at, prosecutors have said.

Czerkawski is in the U.S. on an expired tourist visa and he faces deportation once his case is over in Massachusetts, federal officials have said.

David Traub, a spokesman for Norfolk County District Attorney Michael Morrissey, said Czerkawski faces up to 10 years in prison for the larceny from a person over 60 and up to five years on each of the other three larceny charges. He faces up to five years in prison on each animal cruelty charge.


From 9ba507e076c744f4d394418e4a849e68cd426a4a Mon Sep 17 00:00:00 2001
From: Alex Kube <alexander.j.kube@gmail.com>
Date: Wed, 23 Oct 2019 21:18:56 +0430
Subject: [PATCH 7/9] cmd/go: make GOROOT precious by default

Upstream-Status: Inappropriate [OE specific]

The go build tool normally rebuilds whatever it detects is
stale.  This can be a problem when GOROOT is intended to
be read-only and the go runtime has been built as a shared
library, since we don't want every application to be rebuilding
the shared runtime - particularly in cross-build/packaging
setups, since that would lead to 'abi mismatch' runtime errors.

This patch prevents the install and linkshared actions from
installing to GOROOT unless overridden with the GOROOT_OVERRIDE
environment variable.

Adapted to Go 1.13 from patches originally submitted to
the meta/recipes-devtools/go tree by
Matt Madison <matt@madison.systems>.

Signed-off-by: Alexander J Kube <alexander.j.kube@gmail.com>
---
 src/cmd/go/internal/work/action.go |  3 +++
 src/cmd/go/internal/work/build.go  |  6 ++++++
 src/cmd/go/internal/work/exec.go   | 25 +++++++++++++++++++++++++
 3 files changed, 34 insertions(+)

--- a/src/cmd/go/internal/work/action.go
+++ b/src/cmd/go/internal/work/action.go
@@ -670,6 +670,9 @@ func (b *Builder) addTransitiveLinkDeps(
 			if p1 == nil || p1.Shlib == "" || haveShlib[filepath.Base(p1.Shlib)] {
 				continue
 			}
+			if goRootPrecious && (p1.Standard || p1.Goroot) {
+				continue
+			}
 			haveShlib[filepath.Base(p1.Shlib)] = true
 			// TODO(rsc): The use of ModeInstall here is suspect, but if we only do ModeBuild,
 			// we'll end up building an overall library or executable that depends at runtime
--- a/src/cmd/go/internal/work/build.go
+++ b/src/cmd/go/internal/work/build.go
@@ -167,6 +167,8 @@ See also: go install, go get, go clean.

 const concurrentGCBackendCompilationEnabledByDefault = true

+var goRootPrecious bool = true
+
 func init() {
 	// break init cycle
 	CmdBuild.Run = runBuild
@@ -179,6 +181,10 @@ func init() {

 	AddBuildFlags(CmdBuild, DefaultBuildFlags)
 	AddBuildFlags(CmdInstall, DefaultBuildFlags)
+
+	if x := os.Getenv("GOROOT_OVERRIDE"); x != "" {
+		goRootPrecious = false
+	}
 }

 // Note that flags consulted by other parts of the code
--- a/src/cmd/go/internal/work/exec.go
+++ b/src/cmd/go/internal/work/exec.go
@@ -468,6 +468,23 @@ func (b *Builder) build(a *Action) (err
 		return errors.New("binary-only packages are no longer supported")
 	}

+	if goRootPrecious && (a.Package.Standard || a.Package.Goroot) {
+		_, err := os.Stat(a.Package.Target)
+		if err == nil {
+			a.built = a.Package.Target
+			a.Target = a.Package.Target
+			a.buildID = b.fileHash(a.Package.Target)
+			a.Package.Stale = false
+			a.Package.StaleReason = "GOROOT-resident package"
+			return nil
+		}
+		a.Package.Stale = true
+		a.Package.StaleReason = "missing or invalid GOROOT-resident package"
+		if b.IsCmdList {
+			return nil
+		}
+	}
+
 	if err := b.Mkdir(a.Objdir); err != nil {
 		return err
 	}
@@ -1520,6 +1537,14 @@ func BuildInstallFunc(b *Builder, a *Act
 		return err
 	}

+	if goRootPrecious && a.Package != nil {
+		p := a.Package
+		if p.Standard || p.Goroot {
+			err := fmt.Errorf("attempting to install package %s into read-only GOROOT", p.ImportPath)
+			return err
+		}
+	}
+
 	if err := b.Mkdir(a.Objdir); err != nil {
 		return err
 	}


Two-year clinical outcomes of patients treated with overlapping absorb scaffolds: An analysis of the ABSORB EXTEND single-arm study.
Preclinical data showed that overlapping (OVP) scaffolds might result in delayed healing and strut coverage compared to nonOVP scaffold segments. Furthermore, OVP in patients could result in increased periprocedure myocardial infarction (MI) rate secondary to side branch occlusion; however, little is known whether this may have an impact on long-term clinical outcomes. ABSORB EXTEND is a prospective, single-arm, open-label clinical study in which 812 patients were enrolled at 56 sites. In this study, we compared the immediate and 2-year clinical outcomes of patients with OVP scaffolds (n = 115) to those of patients with nonOVP scaffolds (n = 697). The primary objective was the comparison of major adverse cardiac event (MACE) (cardiac death, MI and ischemic-driven target lesion revascularization [TLR]) and scaffold thrombosis (ST) rates up to 2 years. Baseline clinical and angiographic characteristics were comparable between cohorts except for longer lesions in the OVP patients as expected (16.7 ± 7.3 vs. 11.6 ± 4.4 mm, P < 0.0001), higher lesion complexity (B2) and numerically smaller vessel size. In-hospital, there was a marked increase in MACE in the OVP cohort (7.0 vs. 0.9%, P = 0.002), exclusively driven by a higher rate of periprocedure MI (7.0 vs. 0.9%, P = 0.002). Long-term MACE did not significantly differ between groups (10.4% in the OVP cohort vs. 6.6% in the no-OVP group, P = 0.1) with comparable rates of cardiac death (0.9 vs. 1.2%, P = 1.0) and ischemia-driven TLR (1.7 vs. 2.5%, P = 1.0). Cumulative incidence of MI was higher in the OVP cohort (7.8 vs. 3.0%, P = 0.02). Of note, the rate of MI between hospital discharge and 2-year follow-up was lower in the OVP cohort (0.8 vs. 2.1%, P = 0.04). Cumulative incidence of definite/probable ST was relatively low and comparable between groups (1.8 vs. 1.5%, P = 0.7). In this low-to-moderate complex population treated with the ABSORB scaffold the OVP group showed a higher incidence of periprocedure MI with no immediate or long-term increase in cardiac death, TLR or ST.


A pilot yoga physical education curriculum to promote positive body image.
We examined the effects of a pilot yoga-based physical education (PE) curriculum by testing for change in trait body surveillance, physical self-worth, and body appreciation. Further, we examined the relationships among change in body image variables and the role of state mindfulness in predicting state body surveillance during classes. Adolescents participated in 12 weeks of yoga-based (n=20; Mage=16.45, 90% female) or traditional (n=23;Mage=14.52, 57% female) PE. Results showed significant (p=.004), moderate decreases in trait body surveillance and minimal nonsignificant (p=.11) increases in physical self-worth. Change in trait body surveillance was inversely related to change in physical self-worth and body appreciation in yoga participants. Multi-level modeling analyses revealed that more mindful students also surveyed their body less during class. Intentionally structured yoga participation may support positive body image among adolescents.


A multitask learning model for online pattern recognition.
This paper presents a new learning algorithm for multitask pattern recognition (MTPR) problems. We consider learning multiple multiclass classification tasks online where no information is ever provided about the task category of a training example. The algorithm thus needs an automated task recognition capability to properly learn the different classification tasks. The learning mode is "online" where training examples for different tasks are mixed in a random fashion and given sequentially one after another. We assume that the classification tasks are related to each other and that both the tasks and their training examples appear in random during "online training." Thus, the learning algorithm has to continually switch from learning one task to another whenever the training examples change to a different task. This also implies that the learning algorithm has to detect task changes automatically and utilize knowledge of previous tasks for learning new tasks fast. The performance of the algorithm is evaluated for ten MTPR problems using five University of California at Irvine (UCI) data sets. The experiments verify that the proposed algorithm can indeed acquire and accumulate task knowledge and that the transfer of knowledge from tasks already learned enhances the speed of knowledge acquisition on new tasks and the final classification accuracy. In addition, the task categorization accuracy is greatly improved for all MTPR problems by introducing the reorganization process even if the presentation order of class training examples is fairly biased.


                                      IN THE DISTRICT COURT OF APPEAL
                                      FIRST DISTRICT, STATE OF FLORIDA

JUAN E. RIVAS,                        NOT FINAL UNTIL TIME EXPIRES TO
                                      FILE MOTION FOR REHEARING AND
      Appellant,                      DISPOSITION THEREOF IF FILED

v.                                    CASE NO. 1D14-1440

OASIS OUTSOURCING, INC.
AND SEDGWICK CLAIMS
MANAGEMENT SERVICES,

      Appellees.

_____________________________/

Opinion filed September 23, 2014.

An appeal from an order of the Judge of Compensation Claims.
John J. Lazzara, Judge.

Date of Accident: July 29, 2013.

Paul S. Rosenberg of Rosenberg & Rosenberg, P.A., Hollywood, and Kimberly A.
Hill of Kimberly A. Hill, P.L., Fort Lauderdale, for Appellant.

Rene Lopez, Miramar, for Appellees.

PER CURIAM.

      In this workers’ compensation case, Claimant argues that the Judge of

Compensation Claims (JCC) erred in failing to approve a stipulation whereby the

Employer/Carrier (E/C) would pay Claimant’s attorney a $1,500 attorney’s fee,
under section 440.34(3)(a) and (7), Florida Statutes (2013), and reimburse costs in

the amount of $275. Because it appears the JCC overlooked or ignored facts in

rejecting the stipulation, we reverse and remand for further proceedings.

      Claimant injured his low back and knee in the course and scope of his

employment on July 29, 2013. With the assistance of his attorney, Claimant filed a

petition for benefits (PFB) on September 10, 2013, requesting authorization for a

primary care provider and managed care coordinator as well as orthopedic care. The

E/C filed a response on September 12, 2013, asserting that Claimant was under the

care of an authorized treating orthopedic physician, Dr. Garcia. By notice filed with

the JCC on September 19, the parties agreed to privately mediate the case and

scheduled the mediation for December 13, 2013.

      Nonetheless, Claimant filed a second PFB on November 22, 2013, requesting

authorization   of   Dr.   Weinberg,     D.C.,   as   Claimant’s    managed     care

coordinator/primary care provider. On November 30, 2013, the E/C filed a response,

advising that it would not authorize Dr. Weinberg as Claimant’s managed care

coordinator/primary care physician; that Claimant has such a provider in Dr. Garcia;

that there is a limit on chiropractic care under Florida’s workers’ compensation law;

and that if Claimant was referred for chiropractic care, such care would be

authorized.

                                         2
      A third PFB was filed on the day of the mediation, December 13, 2013,

seeking authorization of an MRI as recommended by Dr. Weinberg in his November

23, 2013, report as well as payment of his bill in the amount of $225 for that date of

service. At the mediation the parties reached an agreement to resolve all of the issues

by settling the case for $10,000, from which Claimant would pay his attorney a fee

of $1,750 and reimburse costs in the amount of $250. In addition, the E/C agreed to

reimburse Claimant $225, the cost of the evaluation with Dr. Weinberg, and to pay

Claimant’s attorney a fee of $1,500 and costs totaling $275.

      The JCC approved the $1,750 Claimant-paid attorney’s fee associated with

the settlement, but denied approval of the $1,500 E/C-paid fee. See § 440.105(3)(c),

Fla. Stat. (2013) (requiring JCC approve all attorney’s fees paid “on account of

services rendered for a person in connection with any proceedings arising under this

chapter”). In the order, the JCC stated he based his findings on a review of “the

stipulation and taking judicial notice of the appropriate relevant pleadings in the

tribunal’s computer file, otherwise known as the ‘docket.’” The JCC found that the

agreed-upon fee was based on a resolution of a claim raised in the PFB filed on

December 13, the day of the mediation; thus, he concluded, there was no evidence

the fee was “earned.” The JCC did advise the parties that they could

      seek reconsideration or modification of this denial by timely motion for
      rehearing, pursuant to Rule 60Q-6.112, Fla. Admin. Code, and
      accompanied with appropriate documentation or explanation
      addressing the above deficiencies. If any party seeks a telephonic re-
                                          3
      hearing on the issue, they may express their request in the motion and
      the extraordinary circumstances which support the request for a
      telephonic re-hearing.

      The parties thereafter jointly filed a motion for approval of the E/C-paid fee

and attached copies of the September and November PFBs and responses to the

PFBs. The parties advised the JCC that the basis for entitlement to the medical-only

fee was not the December PFB; rather, it was the earlier-filed petitions. The parties

again requested the JCC approve the fee, or, in the alternative, schedule a telephonic

hearing should the JCC need further clarification or information.

      In the order rendered in response to the joint motion, the JCC found there were

“no exceptional circumstances or good cause shown in the motion or the docket

necessitating a hearing on the motion.” Further, the JCC concluded that “counsel

for the parties have presented no additional evidence or credible arguments for the

approval of a carrier-paid attorney fee of $1,500.00.” The JCC once again denied

approval of the $1,500 E/C-paid attorney’s fee.

      A JCC’s finding of facts will be upheld on appeal if it is supported by the

record and if it appears that the JCC did not overlook or ignore any

facts. See Chavarria v. Selugal Clothing, Inc., 840 So. 2d 1071, 1078-79 (Fla. 1st

DCA 2003). Section 440.34(3)(a) provides for the payment of an E/C-paid fee if a

claimant successfully asserts a PFB for medical benefits only and has not filed, nor

is entitled to file, a claim for indemnity benefits at that point in time. Section

                                          4
440.34(7) provides that “[i]f an attorney’s fee is owed under paragraph (3)(a), the

[JCC] may approve an alternative attorney’s fee not to exceed $1,500 only once per

accident, based on a maximum hourly rate of $150 per hour.” Here, both parties

allege there were two PFBs filed prior to the December PFB which, like the

December PFB, also requested only medical benefits and both parties requested the

JCC to consider those two PFBs in determining the appropriateness of the $1,500

E/C-paid fee. Because the JCC made no mention of those PFBs in his second order,

we can only conclude that the JCC overlooked or ignored those PFBs. Accordingly,

it is appropriate that we reverse and remand this matter to the JCC for consideration

of those PFBs.

      Based on the foregoing, we REVERSE and REMAND this matter for

proceedings consistent with this opinion.

WOLF, ROWE, and OSTERHAUS, JJ., CONCUR.

                                            5


When Steve Rogers refuses to become a government operative, he is stripped of his uniform, title and shield, and a new Captain America is born! John Walker, the former Superpatriot and future USAgent-along with sidekick Bucky, the future Battle Star-does his best to fill the big shoes Rogers left behind.Collecting: Captain America #332-350, Iron Man #228

From the Publisher:When Steve Rogers refuses to become a government operative, he is stripped of his uniform, title and shield, and a new Captain America is born! John Walker, the former Superpatriot and future USAgent-along with sidekick Bucky, the future Battle Star-does his best to fill the big shoes Rogers left behind.Collecting: Captain America #332-350, Iron Man #228


Why do we Christians tend to treat God as if He was our boss? When you pray do you pray "if it is your will" or "God please" or "I'm begging you God" or "if you do this I'll do that" or "I hope you are not still mad at me" or "I hope this is a good time to ask you for something"? When Jesus prayed He never said please or begged God to answer His prayers! Jesus is our example of what being a Christian is all about. You are to come with boldness before the throne of God. None of the above examples represents any kind of boldness. God is not your boss He is your King! There is a difference between a boss and a king. The worst thing that can happen to you with your boss is that you could lose your job. The worst thing that can happen to you with a king is that you lose your life. Now lets kick it up a notch. For the unsaved, the worst thing that can happen to you with the King of Kings is that not only do you lose your earthly life but that you spend eternity in Hell. You are eternally separated from God in the lake of fire for your unrepented sinful life and you are tormented forever, the second death!

There are those Christians who try to get out of what God has called them to do thinking that they can treat Him like their boss. At work you can call in sick, get hurt on the job, go out on leave of absence, pretend that you don't know how to do it or just blow it off. That doesn't work with God because God knows your heart and knows your reasons both good and bad. I have caught myself praying God if you wish, "If you wish"? God doesn't wish, hope, dream or imagine for He is God. If God wants God gets! God is a sovereign God and King! God only has His own self-imposed limits and is not subject to anyone else's limits and/or restrictions! God is King of kings and creator of all things and deserves to be treated as such. To treat Him otherwise is to insult God.

A member of a kingdom is subject to the rules, decrees, laws, protocols and commands of the King. If you are told to do something for the King you are to do it without question. If you don't you will be punished and/or banished from the kingdom. That is the privilege of being a subject of the kingdom. As a child of the King you are not only to obey the King entirely but you carry the King's authority as an extension of His kingdom wherever you go! You are a child of the Most High God, adopted into and a member of the Royal family, a subject of His Kingdom, an ambassador and citizen of Heaven, a foreigner to this world, an alien without a country here on earth. But most of all you are a representative of God's kingdom on earth.

The question you need to ask yourself isdid I represent God's kingdom well today? That is what God asked me after I got mad at another driver one evening. We were both stopped at a traffic light both side by side at the crosswalk. This guy did his very best to block my view with his tall truck when I was trying to make a right and he was to my left and inching out every time I did. After I finally turned I had all kinds of thoughts going through my head. How I could of turned left over in front of him, or turn around and chased him, or get out of my car and go pound on him and his truck, or worst yet him and his friend get out and pound on me! I pulled into Circle K and started being convicted by the Holy Spirit for dwelling on those thoughts. That is when God asked me"How well did you represent my kingdom today?" Now when I go to sleep at night I ask myself how well did I represent His kingdom today?

The King has a signet ring that he uses when sealing letters, decrees, laws and proclamations. He makes a seal in wax with his ring. Seals were used both to seal the item to prevent tampering, as well as to provide proof that the item was actually from the sender and not a forgery. Thus the saying "seal of approval". As a child of the Most High God, King of Kings and Lord of Lords, God has given you His "seal of approval". But better yet, not only do you carry His seal of approval but God has gone one step further. Jesus, the name above all names, has been given to you to use with all power and authority of heaven that Jesus has as the Son of God and as part of the Trinity. You are an extension and representative of His kingdom and an extension of the authority and power given to Jesus Christ by God the Father who lives in you! John 14:12-14 12I tell you the truth, anyone who has faith in me will do what I have been doing. He will do even greater things than these, because I am going to the Father. 13And I will do whatever you ask in my name, so that the Son may bring glory to the Father. 14You may ask me for anything in my name, and I will do it. It is good to be a child of the King!

My name is Steve and I was born and raised in a good Christian home in Phoenix Arizona. In 2004 God spoke to me to let me know that He would start sharing things with me that I would share with others. Little did I know how powerful and meaningful those things would be! Thank you Lord!!


export MakeInc_cmd=${SRCROOT}/makedefs/MakeInc.cmd
export MakeInc_def=${SRCROOT}/makedefs/MakeInc.def
export MakeInc_rule=${SRCROOT}/makedefs/MakeInc.rule
export MakeInc_dir=${SRCROOT}/makedefs/MakeInc.dir

include $(MakeInc_cmd)
include $(MakeInc_def)

INSTINC_SUBDIRS = pexpert
INSTINC_SUBDIRS_X86_64 = pexpert
INSTINC_SUBDIRS_X86_64H = pexpert
INSTINC_SUBDIRS_ARM = pexpert
INSTINC_SUBDIRS_ARM64 = pexpert

EXPINC_SUBDIRS = pexpert
EXPINC_SUBDIRS_X86_64 = pexpert
EXPINC_SUBDIRS_X86_64H = pexpert
EXPINC_SUBDIRS_ARM = pexpert
EXPINC_SUBDIRS_ARM64 = pexpert

COMP_SUBDIRS =	\
	conf

include $(MakeInc_rule)
include $(MakeInc_dir)


$NetBSD: patch-ad,v 1.3 2001/01/23 20:55:44 jtb Exp $

--- cosqb.f.orig	Fri Nov 17 16:48:23 2000
+++ cosqb.f
@@ -1,5 +1,5 @@
       SUBROUTINE COSQB (N,X,WSAVE)
-      DIMENSION       X(1)       ,WSAVE(1)
+      DIMENSION       X(*)       ,WSAVE(1)
       DATA TSQRT2 /2.82842712474619/
       IF (N-2) 101,102,103
   101 X(1) = 4.*X(1)


---
abstract: 'We investigated the validity of fluctuation-dissipation relations in the non-equilibrium stationary state of fluidized granular media under gravity by two independent approaches, based on theory and numerical simulations. A phenomenological Langevin-type theory describing the fluctuation of center of mass height, which was originally constructed for a one-dimensional granular gas on a vibrating bottom plate, was generalized to any dimensionality, even for the case in which the vibrating bottom plate is replaced by a thermal wall. The theory predicts a fluctuation-dissipation relation known to be satisfied at equilibrium, with a modification that replaces the equilibrium temperature by an effective temperature defined by the center of mass kinetic energy. To test the validity of the fluctuation-dissipation relation, we performed extensive and accurate event-driven molecular dynamics simulations for the model system with a thermal wall at the bottom. The power spectrum and response function of the center of mass height were measured and closely compared with theoretical predictions. It is shown that the fluctuation-dissipation relation for the granular system is satisfied, especially in the high-frequency (short time) region, for a wide range of system parameters. Finally, we describe the relationship between systematic deviations in the low-frequency (long time) region and the time scales of the driven granular system.'
author:
- 'Jun’ichi Wakou$^{1,2}$'
- 'Masaharu Isobe$^{3}$'
title: 'Fluctuation-Dissipation Relations for Motions of Center of Mass in Driven Granular Fluids under Gravity'
---

Introduction
============

Granular materials show fluid-like behavior when they are supplied sufficient energy by external vibration. Fluidized states of granular matter have been studied as interesting examples of non-equilibrium fluids. They exhibit a rich variety of phenomena such as convection, pattern formation on the surface, and segregation (see Ref. [@AransonTsimring-2006] and references therein). Besides these pattern-forming instabilities, the plain stationary state of vibrated granular fluids without complex spatial structures serves as an archetypal example of non-equilibrium stationary states (NESSs). It has been a fundamental goal for many years to find any thermodynamic-like description or to identify the common property of fluctuations in a wide variety of NESSs in nature.

One of the important issues addressed in this paper is the validity of the fluctuation-dissipation relation (FDR) in granular fluids subject to external vibration. An FDR connects the response of an equilibrium system to a small perturbation with the time correlation of spontaneous fluctuations in the system without perturbation. Recently, there has been much interest in how an FDR is violated or should be modified in ageing systems, such as glass, and in NESSs of various systems. (See Refs. [@Cugliandolo-2011; @MarconiPuglisiRondoniVulpani-2008] for recent reviews.)

For granular systems, FDRs have been studied for several situations. Much work has been devoted to the case of freely cooling granular gas, where the gas develops freely without external forces and “cools” as a result of the dissipative nature of the grain interactions. The aim has been to derive a (modified) Green-Kubo relation from which transport coefficients can be calculated [@GoldhirschVanNoije-2000; @DuftyGarzo-2001; @DuftyBrey-2002; @Dufty-BaskaranBrey-2006]. While there is no stationary state for a freely cooling granular gas, a granular gas NESS can be achieved by supplying energy from outside by means of external forcing. A typical experimental means of injecting energy is to shake a container or vibrate a bottom wall (see, e.g., Ref. [@Duran]). In the case where the shaking (or vibrating) is strong enough to inject energy to all grains by frequent collisions with the vibrating wall, the effect of the vibrating wall is often modeled using a thermal bath that couples to every particle. FDRs in such uniformly driven granular systems have been studied in Refs. [@PuglisiBaldassarriLoreto-2002; @Garzo-2004; @PuglisiBldassaryVulpiani-2007; @BuninShokefLevine-2008; @BreyRuiz-Montero-2010]. Puglisi [*et al*]{}. [@PuglisiBaldassarriLoreto-2002] carried out numerical simulations of a model of uniformly driven granular gas and studied FDRs for two different observables. They observed that the FDRs were satisfied if the equilibrium temperature in the FDR for a system at equilibrium was replaced by the granular temperature, defined as the mean-square fluctuation of the grain velocity. Garzó [@Garzo-2004] studied the diffusion of impurities immersed in a granular gas under the influence of uniform driving forces analytically. They showed that a modified form of the Einstein relation, in which the temperature of the gas is replaced by the temperature of the impurity, is violated due to the non-Maxwellian behavior of the impurity velocity distribution function. Bunin [*et al*]{}. [@BuninShokefLevine-2008] analyzed a mean-field model of uniformly driven granular gas and showed that the effective temperature defined by an FDR depends on the frequency. In the case where the shaking (or vibrating) is not strong enough to be regarded as uniform driving, energy injection through a boundary has to be explicitly considered. Brey [*et al*]{}. [@BreyRuiz-Montero-2010] studied the volume fluctuations of a vibrated low-density granular gas confined at the top by a mobile piston numerically. In this system, energy is supplied from the vibrating bottom wall. They discussed the interpretation of an effective temperature defined by requiring the same relation between fluctuations of volume and compressibility as in equilibrium systems. The FDRs and effective temperatures in much denser systems have also been studied by several authors [@MakseKurchan-2002; @BarratColizzaLoreto-2002; @OhernLiuNagel-2004; @DannaMayorBarratLoretoNori-2003]. Among these studies, we refer to an experimental study by D’Anna [*et al*]{}. [@DannaMayorBarratLoretoNori-2003], because their theory based on a Langevin equation formally has the same form as ours, although their experimental setup was very different. They performed an experiment to observe the fluctuating motion of a torsion oscillator immersed in vibration-fluidized granular matter and found that it can be described to first approximation by the formalism for Brownian motion in equilibrium, and an FDR with an effective temperature approximately holds.

We investigated the fluctuating motion of the center of mass (COM) in an NESS of granular matter fluidized by an external energy source located at a bottom wall, under the influence of gravity. Instead of using macroscopic probes such as a piston [@BreyRuiz-Montero-2010] or torsion oscillator [@DannaMayorBarratLoretoNori-2003], we focused on the position of the COM, which is observable using digital high-speed photography in experiments [@WarrHuntleyJacques-1995]. Our major motivation for studying fluctuations of the COM is that a simple (or universal) law might hold as a result of the following properties. First, the fluctuations of macrovariables such as the COM position often possess the largest time scales in the system. Second, they are expected to be Gaussian in a similar sense to the central limit theorem. (In the case of a Markovian stochastic process, the Gaussian property of macrovariables fluctuations can indeed be derived from a master equation of the Markovian process [@vanKampen] .) With this expectation, we proposed a phenomenological theory based on a simple formalism for Brownian motion that describes the motion of the COM height in the NESS of a one-dimensional vibrated granular fluid [@WakouOchiaiIsobe-2008]. We found that the important qualitative features of the dynamics of the COM in event-driven molecular dynamics (MD) simulations were all accounted for by the theory. The theory was extended to a two-dimensional granular fluid on a thermal wall [@WakouIsobe-2010]. Here, we show that when we apply the phenomenological theory to granular fluids in higher dimensions, careful consideration of time scales in granular hydrodynamics [@BreyRuiz-MonteroMoreno-2001; @BrombergLivneMeerson-2003] is necessary. Within the time range for which our theory is valid, it predicts the existence of an FDR. However, the equilibrium temperature in the FDR for an equilibrium system must be modified by the effective temperature of the COM velocity fluctuation. To test our prediction, we performed extensive and accurate event-driven MD simulations for a two-dimensional system of inelastic hard disks on a thermal wall.

Our main result is that an FDR with an effective temperature holds within statistical uncertainty for simulations in a high-frequency (short time) region, while it is violated in a low-frequency (long time) region. The effective temperature is defined by the COM kinetic energy. We observed in our simulations that the ratio between the effective temperature and the global granular temperature increases with inelasticity; the former can be more than four times larger than the latter for the highest inelasticity case.

This paper is organized as follows. In Sec. II, we describe a model granular system and discuss important time scales in the system. In Sec. III, the Langevin equation is introduced, and analytical expressions for the power spectrum and response function of the COM height are described briefly. We also remark on the FDR between these two functions. The complete derivation of the Langevin equation and detailed calculation for the power spectrum and the response function are summarized in Appendix A and B, respectively. A comparison between the theoretical predictions and an extensive event-driven MD are shown in Sec. IV. Finally, in Sec. V, we summarize the main results for FDR validity and comment on the relation between the systematic deviations in the low-frequency (long time) region and the time scales of the driven granular system.

The model system
================

System
------

As a model of grains bouncing on a vibrating bottom plate under gravity, we consider a $d$-dimensional system of $N$ inelastic particles on a “thermal” bottom wall in a constant gravitational field $g$. The particles in the system have diameter $\sigma$ and mass $m$; the total mass of particles is denoted by $M$ ($=Nm$). The thermal wall is kept at a constant temperature $T_0$, which plays the role of a heat source supplying sufficient translational energy to the particles to fluidize them. The $z$-direction is chosen to be opposite to the direction of gravity, and the thermal wall is fixed at $z=0$. For simplicity, we adopt periodic boundary conditions in horizontal directions, so as to ignore the boundary (side-wall) effects. Collisions between particles are inelastic; inelasticity of the particle collisions is characterized by a normal restitution coefficient $r$. To avoid any pattern-forming instability in the horizontal directions, we chose both the inelasticity and linear scales of the system in horizontal directions to be sufficiently small that the system remained homogeneous in the horizontal directions. These conditions are discussed in more detail in Sec. IV.

Time scales
-----------

Before discussing the important time scales in the system, we define several quantities that characterize the macroscopic properties of the system. We first define the kinetic energy per particle as $K(t)\equiv (1/N)\sum_{i=1}^{N} mv_i(t)^2/2$ and the long time average of $K(t)$ in an NESS as $\overline{K}\equiv \lim_{T\to\infty}(1/T)\int_{0}^{T}K(t)dt$ (hereafter, the overline on a quantity represents its long time average in an NESS). We also define the global granular temperature $T$ as $k_B T\equiv (2/d)\overline{K}$, where $k_B$ is the Boltzmann constant, and the thermal velocity as $c\equiv (dk_B T/m)^{1/2}=(2\overline{K}/m)^{1/2}$. A characteristic length scale of the system in the vertical direction $l$ is then defined as $l\equiv c^2/g$.

Bromberg [*et al*]{}. [@BrombergLivneMeerson-2003] have suggested that there are three important time scales in this system at the hydrodynamical level: the macroscopic oscillation time $\tau_{\rm osc}$ (referred to as the “fast time scale” in Ref. [@BrombergLivneMeerson-2003]), the relaxation time for thermal conduction $\tau_{\rm therm}$, and the relaxation time for collisional dissipation $\tau_{\rm diss}$.

For simplicity, we assume that the system is nearly homogeneous, although this is not true for small $r$ and large $N$. The time scale $\tau_{\rm osc}$ represents the period of the slowest oscillation in the vertical direction, that is, the period of the sound mode with the longest wavelength. Thus, $\tau_{\rm osc}\sim l/c_s$, where $c_s$ is the sound velocity. Assuming $c_s\sim c$, which is satisfied for a normal gas, $\tau_{\rm osc}$ can be estimated as $\tau_{\rm osc}\sim c/g$. Because $l/c_s$ also characterizes the pressure relaxation time $\tau_p$, we can regard $\tau_p$ and $\tau_{\rm osc}$ as on the same order, $\tau_p\sim\tau_{\rm osc}\sim c/g$. The relaxation time for thermal conduction $\tau_{\rm therm}$ is estimated as $\tau_{\rm therm}\sim l^2/(\kappa/\rho c_p)$, where $\kappa$ is the thermal conductivity, $\rho$ is the mass density, and $c_p$ is the specific heat at constant pressure [@LandauLifshitzFluid]. $\rho$ can be estimated as $\rho\sim M/(l A)\sim mN_z/(l \sigma^{d-1})$, where $A$ represents the area of the bottom plate in three dimensions ($A$ represents the length of the bottom plate in two dimensions and $A=1$ in one dimension) and $N_z$ represents the number of monolayers at rest. $\kappa$ and $c_p$ are obtained from kinetic theory for elastic spheres and disks [@ChapmanCawling]: $\kappa\sim k_B c/\sigma^{d-1}$ and $c_p\sim k_B/m$. Substituting these results, we obtain $\tau_{\rm therm}\sim N_z c/g$. The relaxation time for collisional dissipation $\tau_{\rm diss}$ can be estimated as the inverse of $(1-r^2)\nu$, where $\nu$ is the collision frequency between two particles. Substituting the lowest order estimation of $\nu$ based on kinetic theory, $\nu\sim \rho \sigma^{d-1} c/m\sim N_z c /l$, we obtain $\tau_{\rm diss}\sim (1-r^2)^{-1}c / (N_z g)$.

The time scales estimated above are summarized as follows: $$\begin{aligned}
&& \hspace{-0.8cm}
\tau_{\rm osc} \sim \tau_{p}\sim \frac{c}{g},
\hspace{0.5cm}
 \tau_{\rm therm}\sim N_z \frac{c}{g},
\nonumber\\
&& \hspace{-0.8cm}
\tau_{\rm diss} \sim \left[N_z(1-r^2)\right]^{-1}\frac{c}{g}.
\label{timescales}\end{aligned}$$ It is important to note that all time scales, $\tau_{\rm osc}$, $\tau_{p}$, $\tau_{\rm therm}$, and $\tau_{\rm diss}$, are proportional to $c/g$. This means that for a system with given $N_z$ and $r$, the macroscopic dynamics with time scaled by $c/g$ are independent of $g$. We utilize this fact later to obtain a frequency response function in an efficient way.

There are three dimensionless parameters, obtained as the ratios between two of these three time scales. The first is $\tau_{\rm therm}/\tau_{\rm osc}\sim N_z$. The second is $\tau_{\rm osc}/\tau_{\rm diss}\sim N_z(1-r^2)$. The third is $\tau_{\rm therm}/\tau_{\rm diss}\sim N_z^2(1-r^2)$. The first and third parameters are the governing parameters for the hydrodynamic description of the system, as introduced by Bromberg [*et al*]{}. [@BrombergLivneMeerson-2003]. They showed that the steady-state profile is governed only by the parameter $$\begin{aligned}
\Lambda\equiv \frac{\sqrt{\pi}}{2}N_z(1-r^2)^{1/2},
\label{lambda}\end{aligned}$$ which is proportional to $(\tau_{\rm therm}/\tau_{\rm diss})^{1/2}$. If $1-r\ll 1$, the second parameter $\tau_{\rm osc}/\tau_{\rm diss}$ is related to $X\equiv N_z(1-r)$. It plays the role of the governing transition parameter from a condensed to fluidized state in a one-dimensional column of beads on a vibrating bottom plate [@LudingClementBlumenRajchenbachDuran-1994]. In our study, we consider the case $N_z\gg 1$ and assume $\tau_{\rm therm}\gg \tau_{\rm osc}$ in the following theoretical analysis.

Theoretical Derivation of the Fluctuation-dissipation Relation
==============================================================

Here, we summarize the theoretical derivation of (i) the power spectrum, (ii) the frequency response function, and (iii) the FDR between (i) and (ii). First, we introduce a Langevin equation as a first approximation that describes the fluctuating motion of the COM on the fast time scales $\tau_{\rm osc}$ and $\tau_p$. Note that the derivation of our theory has already been published in Ref. [@WakouOchiaiIsobe-2008]. We assume $\tau_{\rm therm}\gg\tau_{\rm osc}$, as mentioned above, and focus on the dynamics of the COM on the time scale $\tau_{\rm osc}$, ignoring the significant slow relaxation process of fluctuations of global granular temperature around its stationary value $(2/d)\overline{K}/k_{B}$. The effect of this slow dynamics of granular temperature and validity of our time scale assumption are discussed later.

We summarize the details of the derivation of our Langevin formalism in Appendix A and show the final result here. We denote the height of the COM of granular fluids at time $t$ as $Z(t)$, the time average of $Z(t)$ over a long time interval in an NESS as $\overline{Z}$, and small deviations of $Z(t)$ from $\overline{Z}$ as $\delta Z(t)\equiv Z(t)-\overline{Z}$. The Langevin equation for fluctuating motion of $\delta Z(t)$ is given by (see Eq. (\[langevin1\]) in Appendix A) $$\begin{aligned}
\frac{d^2\delta Z}{dt^2}=-\Omega^2\delta{Z}-\mu\frac{d \delta Z}{dt} + \frac{R(t)}{M},
\label{langevin}\end{aligned}$$ where $R(t)$ represents a random force, which is assumed to be a Gaussian white noise: $$\left<R(t)\right>=0,\quad \quad \left<R(t)R(t')\right>=I\delta(t-t').
\label{gaussianwhite}$$ The brackets $\left<\cdots\right>$ denote an average over the random force. In NESS, it is reasonable to assume $\left<Z(t)\right>_{\rm st}=\overline{Z}$, where $\left<\cdots\right>_{\rm st}$ represents the average in a stationary state. The constant $I$ represents the intensity of the random force, which is related to the second moment of the COM velocity fluctuations. This relation can be obtained by calculating the average kinetic energy of the COM motion in $z$-direction $K_{\rm COMz}\equiv \left< M V_z(t)^2/2\right>_{\rm st}$, where $V_z$ is the $z$-component of the velocity of the COM, $V_z(t)\equiv \frac{d Z(t)}{dt}$. Using an analytical solution Eq. (\[cmheight\]) of the Langevin equation, we obtain $K_{\rm COMz}=I/4M\mu$. Hence, the constant $I$ is identified as $$\begin{aligned}
I\equiv 4M\mu K_{\rm COMz}.
\label{intensity}\end{aligned}$$

This is the same procedure used to determine the noise intensity $I$ when the Langevin equation describes fluctuations in equilibrium at temperature $T$. In equilibrium, equipartition of energy implies $K_{\rm COMz}=k_B T/2$, that is, the mean kinetic energy of the COM in the $z$-direction $K_{\rm COMz}$ and the mean kinetic energy of a particle in one direction $k_B T/2$ are the same. Thus, we obtain the well-known result $I = 2M\mu k_B T$. In the case of an NESS of granular fluids, the violation of equipartition of energy is observed in various systems. A heated binary granular system (see Ref. [@WangMenon-2008] and references therein) is one notable example in which non-equipartition between the mean kinetic energies of two species has been studied. Later, we present numerical simulations that clearly show violation of equipartition, $K_{\rm COMz}\neq k_B T/2$, when we recognize $T$ as the global granular temperature.

The coefficients $\Omega$ and $\mu$ describe an angular frequency of the slowest oscillation of the COM height and frictional coefficient with respect to relative motion of the COM height against the bottom wall, respectively. According to the time scales we consider here, we assume $\Omega\sim \tau_{\rm osc}^{-1}$ and $\mu \sim \tau_{p}^{-1}$ and write them as $$\begin{aligned}
\Omega=\hat{\Omega}g/c,\hspace{0.5cm}
\mu=\hat{\mu} g/c.
\label{coefficients}\end{aligned}$$ Because values of the coefficients $\hat{\Omega}$ and $\hat{\mu}$ cannot be estimated in our phenomenological theory, they are fixed as fitting parameters when we compare results of simulations with the theoretical predictions.

[**Power Spectrum**]{} The power spectrum $S(\omega)$ that represents the fluctuations of $Z$ around the NESS is defined as the Fourier transform of the time correlation function, $$\begin{aligned}
S(\omega)\equiv \int_{-\infty}^{\infty}dt \,e^{-i\omega t}\left<\delta Z(0)\delta Z(t)\right>_{\rm st}.
\label{psdefinition}\end{aligned}$$ The derivation of $S(\omega)$ using the analytic solution of the Langevin equation is straightforward. The final expression of $S(\omega)$ in this system is $$\begin{aligned}
S(\omega)
&=&
\frac{1}{M}\frac{4\mu K_{\rm COMz}}{\left(\Omega^2-\omega^2\right)^2+\left(\mu \omega\right)^2}.
\label{powerspectrum}\end{aligned}$$ See Appendix B for a detailed derivation.

[**Response Function**]{} The frequency response function $\chi(\omega)$ that characterizes the linear response of $Z$ in the NESS against a small external force $\varepsilon f(t)$ can be defined as $$\begin{aligned}
 \chi(\omega)\equiv \lim_{\varepsilon\to 0} \left<\delta \tilde{Z}(\omega)\right> / \varepsilon \tilde{f}(\omega),
\label{resdefinition}\end{aligned}$$ where $\delta \tilde{Z}(\omega)$ and $\tilde{f}(\omega)$ are the Fourier transform of $\delta Z(t)$ and $f(t)$, respectively. The analytical expression of $\chi(\omega)$ is given as $$\begin{aligned}
\chi(\omega)=\frac{1}{M}\frac{1}{\Omega^2-\omega^2+i\mu\omega}.
\label{chi}\end{aligned}$$ A detailed derivation is given in Appendix B. According to conventional definition, $\chi(\omega)$ can be decomposed into real $\chi'(\omega)$ and imaginary $\chi''(\omega)$ parts as $\chi(\omega)=\chi'(\omega)-i\chi''(\omega)$. Thus, we obtain the expression $$\begin{aligned}
\chi'(\omega)&=&\frac{1}{M}\frac{\Omega^2-\omega^2}{(\Omega^2-\omega^2)^2+(\mu\omega)^2},
\label{chi'}\end{aligned}$$ $$\begin{aligned}
\chi''(\omega)&=&\frac{1}{M}\frac{\mu\omega}{(\Omega^2-\omega^2)^2+(\mu\omega)^2}.
\label{chi"}\end{aligned}$$

[**Fluctuation Dissipation Relation**]{} Comparing Eqs. (\[powerspectrum\]) and (\[chi"\]), we obtain the FDR $$\begin{aligned}
\frac{\omega S(\omega)}{2k_B T_{eff}}=\chi''(\omega),
\label{fdt}\end{aligned}$$ where $T_{eff}$ is an effective temperature defined as $T_{eff}\equiv 2K_{\rm COMz}/k_B$. This has the same form as the FDR in an equilibrium system except for $T_{eff}$, which replaces the equilibrium temperature.

Numerical simulations
=====================

Here, we compare the three theoretical predictions described in the previous section with results of the numerical simulation of a two-dimensional granular gas system. The predictions are the power spectrum Eq. (\[powerspectrum\]), the frequency response function Eq. (\[chi\]), and the fluctuation-dissipation relation Eq. (\[fdt\]). Our system consisted of $N$ inelastic hard disks of mass $m$ and diameter $\sigma$ moving in two dimensions on a thermal wall with a fixed temperature $T_0$. Here, the $x$- and $z$-axes represent the horizontal and vertical directions of the system, respectively. The system width is denoted as $L$, and periodic boundary conditions were adopted in the horizontal direction at $x=0$ and $x=L$. The bottom wall was located at $z=0$, and there was no top wall. Gravitational force was exerted on each disk along the negative $z$-direction. Inelastic collisions between hard disks were considered by the normal restitution coefficient $r$. When a disk collided with a thermal wall at the bottom, it left with a value of $z$-component of velocity $v_z$ sampled from the probability density $$\begin{aligned}
p(v_z)=\frac{mv_z}{k_B T_0}\exp \left(-\frac{mv_z^2}{2k_B T_0}\right).\end{aligned}$$ The horizontal component of velocity did not change during the collision.

Numerical simulations were performed with an event-driven algorithm devised to enhance the speed of calculation in dense hard sphere systems [@Isobe-1999]. In the following, all simulation data are presented with mass, length, and time in units of $m$, $\sigma$, and $\sigma/(k_B T_0/m)^{1/2}$, respectively. This corresponds to choosing $k_B T_0=1$. We set $N=5000$ and $L=100$ (these parameters are unchanged throughout this paper). For our main results, $r=0.99- 0.999$ and $g =10^{-3}$ were used unless otherwise mentioned. These correspond to $N_z=50$, $0.05\le X \le 0.5$ and $1.98\le \Lambda<6.25$. A system of width $L=100$ for $r\ge 0.99$ is small enough to prevent any horizontal pattern formation (e.g., ripples or undulations). The global temperature $T$ and the thermal velocity $c$ were calculated using $T=\overline{K}/k_{B}$ and $c=(2\overline{K}/m)^{1/2}$, where $\overline{K}$ is the long time average of the kinetic energy per disk.

Macroscopic properties in the NESS
----------------------------------

In Fig. \[snapshot\] (top), we show typical snapshots of particle configurations in the system of $N=5000$ and $g=10^{-3}$ for $r=0.999$ and $0.992$. The corresponding area-fraction profiles are plotted in Fig. \[snapshot\] (bottom). For a nearly elastic case ($r=0.999$), the profile had one peak around the height $z\simeq 350$. However, the area fraction was relatively dilute (less than $~0.06$), even at the height of the peak. Many inelastic particles were raised up relatively high, like the equilibrium profile of the Boltzmann distribution. In contrast, for $r=0.992$, the profile drastically changed. Most particles condensed at a relatively low level in a cluster; the area-fraction profile showed a clear peak above the low-density region around the thermal wall. This state is known as [*density inversion state*]{} and has been observed in many experiments [@KudrolliWolpertGollub-1997; @WildmanHuntleyHansen-2001] and simulations [@LanRosato-1995; @IsobeNakanishi-1999] of vibrofluidized granular matter.

In accordance with the theoretical study by Bromberg [*et al*]{}. [@BrombergLivneMeerson-2003], which showed that the steady state is characterized by a single parameter $\Lambda$, defined in Eq. (\[lambda\]), we plotted the average kinetic energy per disk $\overline{K}$ and the average kinetic energy of the COM $\overline{K}_{\rm COMz}$ as a function of $\Lambda$ in Fig. \[kineticenergies\]. The statistical error bars with standard deviation were also plotted in all figures throughout the paper. $\Lambda=0$ (that is, $r=1$) corresponds to the equilibrium state in which equipartition of energy $2\overline{K}_{\rm COMz}=\overline{K}=k_B T_0=1$ is satisfied. The factor $2$ comes from the fact that $\overline{K}_{\rm COMz}$ is defined using only the $z$-component of the COM velocity. The horizontal component of the velocity of the COM vanished in our simulations because the horizontal component of disk velocity was unchanged on collision with the bottom wall. While $\overline{K}$ systematically decreased following a power law $\sim \Lambda^{-1.48}$, $\overline{K}_{\rm COMz}$ reached a minimum at $\Lambda\simeq 2$ ($r=0.999$) and increased with $\Lambda$ for $\Lambda > 2$. Fig. \[kineticenergies\] clearly indicates that equipartition of energy breaks down when $\Lambda > 2$ (that is, $2\overline{K}_{\rm COMz}\ne \overline{K}$). Similar behavior in much smaller systems has been reported by us [@WakouIsobe-2010]. In Ref. [@BrombergLivneMeerson-2003], it was shown that the density inversion appears above the threshold $\Lambda_c$ ($\Lambda>\Lambda_c$), where $\Lambda_c\simeq 1.06569$. In the density inversion state, which becomes pronounced for $\Lambda>2$, as shown in Fig. \[snapshot\], a low-density and high-temperature gaseous region near the bottom can cause large fluctuations of the dense cluster on top. Therefore, this violation of the equipartition of energy should be closely connected to development of the density inversion.

![\[kineticenergies\] (Color online) Kinetic energy per particle $\overline{K}$ (circles) and kinetic energy of the COM $\overline{K}_{\rm COMz}$ (squares), plotted versus $\Lambda=\frac{\sqrt{\pi}}{2}50(1-r^2)^{1/2}$ for $r=0.9999,\,0.9996,\,0.999,\,0.998,\,0.996,\,0.994,\,0.992$, and $0.99$ from left to right. The solid line gives a numerical fit of the form $1.04\times\Lambda^{-1.48}$. ](./fig2.eps){width="0.8\columnwidth"}

The relation between the long time average of the COM height $\overline{Z}$ and the kinetic energy per particle is given in Fig. 3. A linear relation $\overline{Z}=\overline{K}/mg+\mbox{const.}$ was satisfied for $\overline{K}>0.1$, even when the system had a density inversion with a relatively high density cluster. In an equilibrium system of dilute gases, the relation $\overline{Z}=\overline{K}/mg+\mbox{const.}$ holds as a result of statistical mechanics. The fact that $\overline{K}$ characterizes $\overline{Z}$ in the same way as in equilibrium suggests that the global granular temperature $T$ in the *inhomogeneous* non-equilibrium state still retains the same meaning as the equilibrium temperature, at least in a macroscopic sense.

![\[kin-pot\] (Color online) The average height of the center of mass $\overline{Z}$ versus $\overline{K}$ for $r=0.999,\,0.998,\,0.996,\,0.994,\,0.992$, and $0.99$ from right to left. The error bars are smaller than the size of the marks. The solid line gives a linear fit with the slope $(mg)^{-1}$, where $m=1$ and $g=10^{-3}$. ](./fig3.eps){width="0.8\columnwidth"}

In Fig. \[kin-G\], $\overline{K}$ is plotted as a function of the gravitational acceleration $g$. The dependence of $\overline{K}$ on $g$ turned out to be rather weak. We utilized this fact to measure the response function from simulations in an efficient way (see Sec. IV C).

![\[kin-G\] (Color online) Kinetic energy per particle $\overline{K}$ as a function of $g$. The error bars are smaller than the sizes of the marks. ](./fig4.eps){width="0.8\columnwidth"}

In Fig. \[vdistrib\], we plotted the probability distribution $P(C)$ of the scaled COM velocity $C\equiv V_z/(2 \overline{K}_{\rm COMz}/M)^{1/2}$. The data were fitted sufficiently by a Gaussian for all cases studied in this paper, as expected from the central limit theorem. This Gaussian property was consistent with our theory based on a linear Langevin equation with additive Gaussian noise.

![\[vdistrib\] (Color online) Probability distribution of the scaled COM velocity $C\equiv V_z/(2 \overline{K}_{\rm COMz}/M)^{1/2}$. The solid line is Gaussian with unity dispersion. ](./fig5.eps){width="0.8\columnwidth"}

Power spectrum of the COM height
--------------------------------

We first tested the theoretical prediction Eq. (\[powerspectrum\]) for the power spectrum of the COM height. Using the relation Eq. (\[coefficients\]), Eq. (\[powerspectrum\]) can be rewritten as $$\begin{aligned}
\hat{S}(\hat{\omega})
&\equiv&
S(\hat{\omega} g/c)/\left[ 4\left(\frac{c}{g}\right)^3 \frac{\overline{K}_{\rm COMz}}{M}\right]
\nonumber\\
&=&
\frac{\hat{\mu}}{\left(\hat{\Omega}^2-\hat{\omega}^2\right)^2+\left(\hat{\mu}\hat{\omega}\right)^2},
\label{scps}\end{aligned}$$ where $\hat{\omega}$ is the scaled angular frequency, defined by $\hat{\omega}\equiv \omega c/g$. This expression suggests that if we scale the power spectrum and the angular frequency as in Eq. (\[scps\]), it shows a universal behavior independent of any system parameters.

In Fig. \[pscom\] (top), the power spectrum $S(\omega)$ is plotted for different values of $r$. Two sharp peaks were observed; one is near zero angular frequency ($\omega=0$), the other one is at the angular frequency of the macroscopic oscillation ($\omega=\omega_{\rm osc}$), which increased as $r$ decreased. The heights of both these peaks decreased with $r$. Figure \[pscom\] (bottom) shows the scaled power spectrum $\hat{S}(\hat{\omega})$ obtained by scaling $S(\omega)$ in Fig. \[pscom\] (top), according to Eq. (\[scps\]) using $c$ and $\overline{K}_{\rm COMz}$ calculated from simulation data. The theoretical prediction Eq. (\[scps\]) with fitting numerical parameters $\hat{\mu}=0.50$, $\hat{\Omega}=1.7$ is presented as a thick solid line. It is consistent with the results of simulations for the range $0.99\le r\le 0.996$ in this region near the peak at $\hat{\omega}=\hat{\omega}_{\rm osc}\equiv\omega_{\rm osc}c/g$, where we expect our theory to serve as a first-order approximation. We found large deviations from the theoretical prediction in the region $\hat{\omega}<\hat{\omega}_{\rm osc}$ (the sharp peak near $\hat{\omega}=0$). As we illustrate below, the peak near $\hat{\omega}=0$ could be associated with slow fluctuations of global granular temperature due to thermal conduction and collisional dissipation. Because $\tau_{\rm therm}/\tau_{\rm osc}\sim N_z\gg 1$ and $\tau_{\rm diss}/\tau_{\rm osc}\sim [N_z(1-r^2)]^{-1}\ge 1.0$ for our simulations with $N_z=50$ and $r\ge 0.99$, the contributions of these two processes should appear at $\hat{\omega}<\hat{\omega}_{\rm osc}$. We also found that for $r\ge 0.998$, the simulation data in Fig. \[pscom\] (bottom) deviated from our theory, even in the region near the peak at $\hat{\omega}=\hat{\omega}_{\rm osc}$. These deviations near $\hat{\omega}_{\rm osc}$ could be attributed to the drastic change in density profiles shown in Fig. \[snapshot\] as $r$ is varied. Concerning our theory, the change in density profiles may affect the numerical coefficients $\hat{\Omega}$ and $\hat{\mu}$ in Eq. (\[coefficients\]). Furthermore, in the region $\hat{\omega}<\hat{\omega}_{\rm osc}$, the effect of global temperature fluctuations mentioned above could become pronounced for $r\ge 0.998$, because both $\tau_{\rm therm}$ and $\tau_{\rm diss}$ became much larger than $\tau_{\rm osc}$, and hence the fluctuations had long lifetimes. Nonetheless, a satisfactory explanation of these deviations for $r\ge 0.998$ has not yet been given.

Here, we show simulations suggesting that the behavior of $S(\omega)$ in the region near $\omega =0$ can be described by taking into account the slow dynamics of $K(t)$. We denote the slowly varying part of $K(t)$ as $K'(t)$ and suppose it fluctuates on a much longer time scale than $\tau_{\rm osc}$, due to thermal conduction and collisional dissipation. Then, $K'(t)/k_{B}$ can be regarded as a time-dependent global granular temperature. Similarly, we let $Z'(t)$ denote the slowly varying part of $Z(t)$ on the same time scale as $K'(t)$. We assume here that in this long time scale, $K'(t)$ and $Z'(t)$ play the same role as their long time averages $\overline{K}$ and $\overline{Z}$. That is, they satisfy the same linear relation as their long time averages observed in Fig. \[kin-pot\]: $Z'(t)=K'(t)/mg+\mbox{const.}$ with the same constant factor. If this is the case, the power spectrum of $\delta Z(t)$, $S(\omega)$, in the region near $\omega =0$ should be given by the power spectrum of $\delta K'(t)/mg$, where $\delta K'(t)=K'(t)-\overline{K}$. In Fig. \[pskin\], we show the power spectrum of $\delta K(t)/mg$, where $\delta K(t)=K(t)-\overline{K}$, and $S(\omega)$ for $r=0.999$ and $0.992$. The figure shows that the curves around the peak in $S(\omega)$ near $\omega =0$ and the peak in the power spectrum of $\delta K(t)/mg$ near $\omega =0$ are consistent. The consistency between the two curves is also observed for the other $r$ values. This result indicates that the peak in $S(\omega)$ near $\omega =0$ can be accounted for by slow dynamics of $K(t)$ due to thermal conduction and collisional dissipation. It should be emphasized that in our present theory, fluctuations of granular temperature in both space and time are ignored and only the global granular temperature $T$ is defined, using the long time average of $K(t)$. Further investigation is necessary to construct a theory that fully describes the behavior of $S(\omega)$, taking into account the effect of slow fluctuations of granular temperature.

Response functions
------------------

Next, we test the theoretical prediction Eqs. (\[chi’\]) and (\[chi"\]) for the frequency response functions of the COM. By scaling these functions in the same way as the power spectrum, we can derive universal equations $$\begin{aligned}
\hat{\chi}'(\hat{\omega})
\equiv
\chi'(\hat{\omega} g/c)Mg^2/c^2
=\frac{\hat{\Omega}^2-\hat{\omega}^2}{\left(\hat{\Omega}^2-\hat{\omega}^2\right)^2+\left(\hat{\mu}\hat{\omega}\right)^2},
\label{scaledchireal}\end{aligned}$$ $$\begin{aligned}
\hat{\chi}''(\hat{\omega})
\equiv
\chi''(\hat{\omega}g/c)Mg^2/c^2
=\frac{\hat{\mu}\hat{\omega}}{\left(\hat{\Omega}^2-\hat{\omega}^2\right)^2+\left(\hat{\mu}\hat{\omega}\right)^2}.
\label{scaledchiimag}\end{aligned}$$

The frequency response function was measured using numerical simulations via the following procedure. First, we prepared for a system in the stationary state with a given $N_z$, $r$, and $g$ after a sufficiently long relaxation time from the initial state of particles with randomly distributed positions and velocities. At $t=0$, we exerted a small constant external force on all particles in the direction of gravity and measured the height of the COM at $t>0$; from this COM relaxation process, we deduced a response function by the standard procedure given in textbooks (see, e.g., Ref. [@KuboTodaHashitsume]). In other words, we measured a response function against a step functional external force. The frequency response function was obtained as the Fourier transform of the response function.

It is important to note that in the response of the COM height against a small but finite external force in our system, *nonlinear* effects resulting from time scale changes were non-negligible. This can be seen from the fact that the relevant time scales shown in Eq. (\[timescales\]) all depended on $g$ and that exerting a constant force in the direction of gravity was equivalent to changing $g$. Therefore, a linear response could be defined only in the limit of small external force. This shows that our Langevin-type theory is different from the well-known Langevin theory for Brownian motion in a fixed harmonic potential, where the response of a Brownian particle is linear against a finite external force. Consequently, we had to exert an external force that was much smaller than the gravitational force in our system, in order to measure the linear response of the COM height. Because the fluctuation of the COM height of $5000$ particles was typically much larger than the response against such a small constant force, we needed to perform the response function measurement for a large number of systems with the same parameters $N_z$, $r$, and $g$ but different initial conditions and take an average of the response functions over all realizations. As shown later, in the case of a constant force that is $1\%$ of the gravitational force, we needed more than $10^4$ realizations to obtain sufficient statistics for clear response functions. This required relatively long CPU times that impeded long simulations with a wide range of parameters $r$, $N_z$, and $g$.

We therefore optimized the method by choosing an appropriate parameter to approximately evaluate the response function from a small number of realizations, which could be provided in an acceptable time with our computational facilities. Suppose a system with gravitational field $g$ is initially in an NESS and the gravitational acceleration is increased at $t=0$ from $g$ to $g+\Delta g$. This is equivalent to exerting a step function external force $-M\Delta g\theta(t)$ on the COM height, where $\theta(t)$ is the Heaviside unit step function. Now we define the function $\chi(t;g,g+\Delta g)$ as $$\begin{aligned}
 \chi(t;g,g+\Delta g)\equiv -\frac{d\left<\delta Z\right>_{t}}{dt}/M\Delta g,
\label{chideltag}\end{aligned}$$ where $\left<\cdots\right>_{t}$ represents the average taken over the ensemble of realizations $\delta Z$ at time $t$. This is a function of $\Delta g$ in our system due to the nonlinear effects mentioned above; it would equal the response function only if $\left<\delta Z\right>_t$ were linear in $\Delta g$. We denote the Fourier transform of $\chi(t;g,g+\Delta g)$ as $\chi(\omega;g,g+\Delta g)$. According to Eq. (\[resdefinition\]), the frequency response function $\chi(\omega;g)$ for the system in the stationary state with $g$ is given by $$\begin{aligned}
 \chi(\omega;g)=\lim_{\Delta g\to 0}\chi(\omega;g,g+\Delta g).
\label{chiomegag}\end{aligned}$$

We now consider the time scales that we introduced in Sec. II B, which characterize macroscopic dynamics at $t>0$. As we discussed in Sec. II B, all these time scales in the NESS depend on $g$ in the form $\tau= c(g)/g\times \mbox{const.}$, where we wrote the $g$-dependence of $c$ explicitly for the sake of clarity. Based on our observations in Fig. \[kin-G\] that $c(g)$ changed a few percent as $g$ was increased by 10%, we assumed that the thermal velocity at $t>0$ is given by $c(g)$ if $\Delta g$ is sufficiently small. Thus, these time scales at $t>0$ have the form $\tau= c(g)/(g+\Delta g) \times \mbox{const.}$, where $g+\Delta g$ is the gravitational acceleration at $t>0$. This dependence of all the characteristic time scales on $\Delta g$ leads us to the scaling relation $$\begin{aligned}
 \chi(\omega;g, g+\Delta g)=\frac{1}{M}\left(\frac{c(g)}{g+\Delta g}\right)^2 \hat{\chi}\left(\omega \frac{c(g)}{g+\Delta g}\right),
\label{scaling}\end{aligned}$$ where $\hat{\chi}$ is a non-dimensional function.

As long as Eq. (\[scaling\]) holds, we can estimate the limit in Eq. (\[chiomegag\]) as $$\begin{aligned}
 \chi(\omega;g)=\left(\frac{g+\Delta g}{g}\right)^2
\chi\left(\omega\frac{g+\Delta g}{g}; g, g+\Delta g\right).
\label{chi-scaled}\end{aligned}$$

To verify the validity of the scaling relation Eq. (\[scaling\]), we performed two series of simulations for $r=0.992$. First, we measured the function $\chi(\omega; g,\,g+\Delta g)$ in Eq. (\[chideltag\]) for $\Delta g/g=10^{-2}$, taking the average over $41000$ realizations. Second, we measured the frequency response function $\chi(\omega;g)$ using Eq. (\[chi-scaled\]) for $\Delta g/g=10^{-1}$, taking the average over $800$ realizations. In Figs. \[chi-r0992\], we compare the frequency response functions obtained from these two series of simulations. We found that they were consistent, although there were some discrepancies in $\chi'(\omega)$ near $\omega=0$.

More evidence of validity of the scaling relation comes from the fact that an FDR in an equilibrium system is satisfied when we measured the frequency response function using Eq. (\[chi-scaled\]). This is discussed further later (see Fig. \[fdr-r1\]).

The frequency response functions presented below were obtained using the scaling relation Eq. (\[scaling\]) (and Eq. (\[chi-scaled\])) by averaging over $800$ realizations. In Figs. \[sc-chi\], we show the real (top) and imaginary (bottom) parts of the scaled response functions $\hat{\chi}'$ and $\hat{\chi}''$ as functions of $\hat\omega$. Here, values of $c$ in Eq. (\[scaledchireal\]) and (\[scaledchiimag\]) were calculated in an NESS without perturbation.

Theoretical predictions Eqs. (\[scaledchireal\]) and (\[scaledchiimag\]) with the same (universal) fitting parameter as estimated in Fig. 6, $\hat{\mu}=0.50$ and $\hat{\Omega}=1.7$, are shown by thick lines. It appears that $\hat{\chi}(\omega)$ is consistent with the theoretical predictions if $r\le 0.996$.

Fluctuation-dissipation relation
--------------------------------

To test the FDR Eq. (\[fdt\]) predicted by our theory, we evaluate the left- and right-hand sides independently using the results of simulations on $S(\omega)$ (Sec. II B) and $\chi''(\omega)$ (Sec. II C) presented in previous subsections. Note that $\overline{K}_{\rm COMz}$ were measured in the NESS where $S(\omega)$ was measured.

First, we confirmed that the FDR held within the error bounds of the simulation result in the whole range of $\omega$ given $r=1$ in Fig. \[fdr-r1\]. The stationary state is just the equilibrium state of elastic particles on a thermal wall. In Fig. \[fdr-r0999-r099\], the left- and right-hand sides of the FDR are plotted as a function of $\omega$ for different $r$ values. For all $r$ ($0.99 \le r \le 0.999$), we found that the FDR held within the error bounds in the higher frequency range of $\omega$, including a region near the highest peak at $\omega=\omega_p$. The angular frequency of the highest peak $\omega_p$ was close to $\omega_{\rm osc}$, defined as the angular frequency of a peak in $S(\omega)$. We stress here that we defined $T_{eff}$ as $T_{eff}=2\overline{K}_{\rm COMz}/k_B$ in the FDR. The quantitative agreement in Fig. \[fdr-r0999-r099\] supports this definition of $T_{eff}$, using $\overline{K}_{\rm COMz}$ instead of using the global granular temperature $T$, because $T_{eff}$ is more than three times larger than $T$ for $r\le 0.996$ (see Fig. \[kineticenergies\]).

We found systematic deviations in the region $\omega < \omega_{p}$ for $r=0.999$ and $r\le 0.994$. These deviations were related to the fact that there was a peak near $\omega = 0$ in $S(\omega)$ (shown in Fig. \[pscom\]), while no corresponding peak near $\omega = 0$ appeared in $\chi''(\omega)$. As we discussed in Sec. IV B, the peak near $\omega= 0$ could have been connected with slow fluctuations of granular temperature due to thermal conduction and collisional dissipation. For $r=0.999$, the time scales of these two processes ($\tau_{\rm therm}$ and $\tau_{\rm diss}$) became much larger than $\tau_{\rm osc}$. Hence, the fluctuations with long lifetimes might be responsible for the deviations at small $\omega$. For $r\le 0.994$, the deviation appeared to increase as $r$ decreased. Because the system had lower granular temperature for smaller $r$, a larger heat current from the thermal wall was induced, causing larger fluctuations in global granular temperature. Further investigation is necessary to understand this violation of the FDR more precisely.

![\[fdr-r1\] (Color online) Left-hand side $\omega S(\omega)/2k_B T_{eff}$ and right-hand side $\chi''(\omega)$ of Eq. (\[fdt\]) for $r=1$. $N=5000$, $L=100$, and $g=10^{-2}$, with averages over 400 realizations for $S(\omega)$ and over 800 realizations for $\chi''(\omega)$. ](./fig10.eps){width="0.8\columnwidth"}

![\[fdr-r0999-r099\] (Color online) Left-hand side $\omega S(\omega)/2k_B T_{eff}$ and right-hand side $\chi''(\omega)$ of Eq. (\[fdt\]) for (a) $r=0.999$, (b) $r=0.998$, (c) $r=0.996$, (d) $r=0.994$, (e) $r=0.992$, and (f) $r=0.99$. $N=5000$, $L=100$, and $g=10^{-3}$, with averages over 400 realizations for $S(\omega)$ and over 800 realizations for $\chi''(\omega)$. ](./fig11.eps){width="1.0\columnwidth"}

Conclusion
==========

We studied the validity of the fluctuation-dissipation relation with regard to the COM motion in an NESS of a driven granular fluid under gravity. By neglecting the fluctuations of global temperature caused by thermal conduction and collisional dissipation, which change much slower than the macroscopic oscillation of the fluid, we derived a Langevin equation for the COM height using phenomenological considerations. This equation predicts functional forms of the correlation and response functions for the COM height that contain two phenomenological numerical constants $\hat{\mu}$ and $\hat{\Omega}$, which are used as fitting parameters. It also gives a fluctuation-dissipation relation accompanied by an effective temperature $T_{eff}$ that characterizes the agitating motion of the COM height by $T_{eff}=2\overline{K}_{\rm COMz}/k_B$.

To test the fluctuation-dissipation relation, we performed event-driven MD simulations and measured the power spectrum and response function for the COM height. While the power spectrum was consistent with our theory for $r\le 0.996$ and $\omega$ around the angular frequency of the slowest oscillation of the COM, it also showed large deviations from the theoretical predictions near $\omega=0$ for all $r$ ($0.99\le r\le 0.999$) and in the whole range of $\omega$ for $r> 0.996$. The response function agreed closely with our theory for $r\le 0.996$ but showed deviations for $r> 0.996$. Furthermore, we compared the left- and right-hand sides of the FDR. The results showed that the FDR held in a region of $\omega$ near the highest peak for all cases of $r$ we tested. It was violated near $\omega=0$ for small $r$, $r\le 0.994$, and for $r$ close to unity, $r=0.999$. For $r\le 0.994$, the violation became more pronounced as $r$ decreased. The violation of the FDR was attributed to a peak near $\omega=0$ in the power spectrum for the COM height, which was absent in the imaginary part of the frequency response function. The peak near $\omega=0$ in the power spectrum cannot be described by our theory.

We showed that these deviations near $\omega=0$ could be attributed to slow fluctuations of global temperature, defined as the slowly varying part of the kinetic energy per particle $K(t)$ due to thermal conduction and collisional dissipation. These fluctuations of global temperature were neglected in our theory. The deviations in the power spectrum and resulting violation of the FDR are expected to be accounted for by a theory that describes both $Z(t)$ and $K(t)$, which we will investigate in the future.

In Ref. [@HaradaSasa-2005] a formula that connects the violation of the FDR in an NESS with the energy dissipation, or equivalently the energy input from outside, was proposed. A theory extended to include the slow dynamics of $K(t)$ and direct measurement of energy input in our simulations might give some insight into the generality of their formula.

Finally, the basic question of whether the effective temperature $T_{eff}$ obtained here has any physical meaning in terms of thermodynamics remains. The definition of effective temperature in a system that relaxes in several time scales, typically glass, has been debated in Refs. [@CugliandoroKurchanPeliti-1997; @BerthierBarrat-2002; @Cugliandolo-2011]. It would be interesting to apply their theories to our problem with three time scales $\tau_{\rm therm}$, $\tau_{\rm diss}$, and $\tau_{\rm osc}$. It would also be interesting to investigate via simulation what happens if two systems with different effective temperatures are in contact with each other. Measuring the direction of heat flow directly might clarify the physical meaning of the effective temperature.

J. W. is grateful to H. Nakanishi, T. Sakaue, T. Saito, and C. Nakajima for their hospitality during his stay at Kyushu University, where part of this study was done. This study was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology No. 23740293. Some of the computations for this study were performed using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, and the Research Center for Computational Science (RCCS) in Okazaki, Japan.

Langevin equation
=================

In this section, we summarize the derivation [@WakouOchiaiIsobe-2008; @WakouIsobe-2010] of the Langevin equation that describes the motion of the COM of grains.

The equation of motion for the COM of the grains in the model described in Sec. II can be written as $$M\frac{d^2 Z}{dt^2}=-Mg+F_b.
\label{eqmotion}$$ The right-hand side of the equation of motion for the COM must be, in general, the sum of the external forces acting on the grains. In our model, these are the gravitational force $-Mg$ and the $z$-component of the force exerted by the bottom wall $F_b$. Thus, it is essential to understand the properties of $F_b$ for the study of COM motion.

Let us consider the reaction force $F'_b(=-F_b)$: the force exerted by grains against the bottom wall. A snapshot of the granular fluid is sketched in Fig. \[sketch\] (a). Suppose the COM is at a height $Z$ and is moving downward at velocity $V$. We now change the frame of reference to the center of mass frame (see Fig. \[sketch\] \[b\]); the bottom wall that lies a distance $Z$ away from the COM in the $z$-direction is moving upward with velocity $-V$. Now the problem is how to determine $F'_b$, the force acting on the bottom wall as a result of frequent collisions of granular particles, in the situation shown in Fig. \[sketch\] (b). There, the bottom wall is moving upward with velocity $-V$ against the macroscopically static fluid.

![\[sketch\] (a) Schematic of the system observed in the laboratory frame of reference, (b) the same system observed in the center of mass frame, (c) the Rayleigh piston: a piston that undergoes random collisions with a one-dimensional heat bath of particles. ](fig12.eps){width="8.0cm"}

This problem is similar to the problem of determining the force acting on a one-dimensional Brownian particle (the Rayleigh piston [@vanKampen]) moving in the $z$-direction with a velocity $-V$ (see Fig. \[sketch\] \[c\]). We create an expression for $F'_b$ on the basis of this analogy and assume that $F'_b$ consists of three components. The first is a systematic force $f_P(t)$ that equals the pressure multiplied by the area of the bottom wall. Because the local density near the bottom wall changes according to the motion of the COM, this force may depend on time. Apparently, the long time average of $f_P(t)$, that is, $\overline{f_P}$, must be equal to $-Mg$, the gravitational force acting on all particles. The simplest assumption for the time-dependent part of $f_P(t)$ is that it is proportional to the deviation of the COM height from its stationary value, $Z(t)-\overline{Z}$. This is because the change in local density near the bottom wall is proportional to $-(Z(t)-\overline{Z})$ if the change in the height of the COM is sufficiently small: $|Z(t)-\overline{Z}|/\overline{Z}\ll 1$. The second component is a frictional force. We assume here the simplest form of the frictional force: linear in the relative velocity $-V(t)$ of the bottom wall to the COM. The third component is a random force. We assume $$\begin{aligned}
 F'_b(t)&=&-Mg + M\Omega^2\left(Z(t)-\overline{Z}\right)+M\mu V(t)+R'(t)
\nonumber\\
&=&-F_{b}(t),
\label{fbdash}\end{aligned}$$ where $\Omega$ is a coefficient that specifies the angular frequency of the slowest oscillation of the COM, and $\mu$ is the frictional coefficient. According to the discussion of characteristic time scales in Sec. II B, the time scales for macroscopic oscillation $\tau_{\rm osc}$ and that for pressure relaxation $\tau_{p}$ are $\tau_{\rm osc}\sim\tau_{p}\sim c/g$. Thus, we assume $$\begin{aligned}
\Omega=\hat{\Omega}/\tau_{\rm osc}=\hat{\Omega}g/c,\hspace{0.5cm}
\mu=\hat{\mu}/\tau_{p}=\hat{\mu} g/c.\end{aligned}$$

For the random force, we assume stationary Gaussian white noise in the same way as for the Rayleigh piston: $$\left<R'(t)\right>=0,\quad \quad \left<R'(t)R'(t')\right>=I\delta(t-t').
\label{gaussianwhite2}$$ where $I$ represents the intensity of the random force.

Substituting the $F_b$ obtained in (\[fbdash\]) into the equation of motion of the COM (\[eqmotion\]), we obtain $$\begin{aligned}
\frac{d^2\delta Z}{dt^2}=-\Omega^2\delta{Z}-\mu\frac{d \delta Z}{dt} + \frac{R(t)}{M},
\label{langevin1}\end{aligned}$$ where $\delta Z\equiv Z(t)-\overline{Z}$ and $R(t)=-R'(t)$. The random force $R(t)$ has exactly the same property described in (\[gaussianwhite2\]) as $R'(t)$. Note that the Langevin equation (\[langevin1\]) has the same form as that describing Brownian motion in a harmonic potential.

Derivation of the power spectrum and the response function
==========================================================

Derivation of the power spectrum and the response function from the Langevin equation describing Brownian motion in a harmonic potential is given in textbooks (see e.g., Ref. [@ResiboisLeener]). We therefore present only essential steps in their calculation. First, we consider the power spectrum of the fluctuating motion of the COM obeying the Langevin equation (\[langevin1\]). The formal solution of Eq. (\[langevin1\]) is written as $$\begin{aligned}
Z(t)-\overline{Z}  =
\int_{-\infty}^{t}G(t-t')\frac{R(t')}{M}dt' +F_{ini}(t),
\label{cmheight}\end{aligned}$$ where the function $G(t)$ is given by $$G(t) = \frac{e^{-\frac{\mu}{2}t}}{\omega_0} \sin\left(\omega_0 t\right),
\label{gt}$$ and $\omega_0$ is defined by $\omega_0\equiv(\Omega^2-(\mu/2)^2)^{1/2}$. The last term $F_{ini}(t)$ in Eq. (\[cmheight\]) consists of those that depend on the initial conditions and vanish after a sufficient amount of time. Thus, the term is negligible when calculating long time averages of physical quantities in the stationary state.

Using this formal solution, we can calculate the two-time correlation function $\phi(t)$ in an NESS defined by $\phi(t)\equiv\lim_{t'\to \infty} \left<\delta Z(t')\delta Z(t'+t)\right>$, where the brackets $\left< \cdots\right>$ indicate an average over the random force $R(t)$. We took the limit $t'\to \infty$ to ensure that the system is in the stationary state.

The power spectrum of $\delta Z(t)$ can be obtained using the Winner-Khinchin theorem: $$\begin{aligned}
S(\omega)&=&\int^{\infty}_{-\infty} dt e^{-i\omega t} \phi (t)
\\
&=&
\frac{I}{M^2}\frac{1}{\left(\Omega^2-\omega^2\right)^2+\left(\mu \omega\right)^2}.\end{aligned}$$

Next, we consider the response function for the COM, which describes the linear response of the COM with regard to a small external force $\varepsilon f(t)$. The Langevin equation in this case is written as $$\begin{aligned}
\frac{d^2\delta Z}{dt^2}+\Omega^2\delta{Z}+\mu\frac{d \delta Z}{dt} - \frac{\varepsilon f(t)}{M} - \frac{R(t)}{M} =0.
\label{langevin2}\end{aligned}$$ Taking the average over the random force, we obtain $$\begin{aligned}
\frac{d^2\left<\delta Z\right>}{dt^2}+\Omega^2\left<\delta{Z}\right>+\mu\frac{d \left<\delta Z\right>}{dt}
- \frac{\varepsilon f(t)}{M} =0.
\label{langevin-average}\end{aligned}$$

The response function $\chi (t)$ is defined as $$\begin{aligned}
\left<\delta Z(t) \right>=\int^{t}_{-\infty} dt' \chi(t-t') \varepsilon f(t').
\label{linearresponse3}\end{aligned}$$ Here, the external force $\varepsilon f(t)$ is assumed to be infinitely small. The Fourier transform of this relation yields $\langle \delta \tilde{Z}(\omega) \rangle=\chi(\omega) \varepsilon \tilde{f}(\omega)$, and hence $$\begin{aligned}
\chi(\omega)=\lim_{\varepsilon\to 0}\left<\delta \tilde{Z}(\omega) \right>/ \varepsilon \tilde{f}(\omega).
\label{linearresponse2}\end{aligned}$$

Performing the Fourier transform of the relation (\[langevin-average\]) and comparing it with Eq. (\[linearresponse2\]), we obtain the frequency response function (complex admittance) $$\begin{aligned}
\chi(\omega)=\frac{1}{M}\frac{1}{\Omega^2-\omega^2+i\mu\omega}.\end{aligned}$$

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