# An Order-Complexity Aesthetic Assessment Model for Aesthetic-aware Music
Recommendation

Xin Jin, Wu Zhou, Jinyu Wang  Department of Cyber Security, Beijing Electronic
Science and Technology Institute, Beijing, 100070, ChinaBeijing Institute for
General Artificial Intelligence, Beijing, 100085, China<EMAIL_ADDRESS>,
Duo Xu  Tianjin Conservatory of Music, Tianjin, 300171, ChinaQingdao Academy
of Intelligent Industries, Qingdao, 266114, China
Tongji University, Shanghai, 200082, China<EMAIL_ADDRESS>and  Yongsen Zheng
School of Computer Science and Engineering, Sun Yat-sen University, No. 132,
East Outer Ring Road, Guangzhou-College-Town, Guangzhou, 510006,
<EMAIL_ADDRESS>
(2023)

###### Abstract.

Computational aesthetic evaluation has made remarkable contribution to visual
art works, but its application to music is still rare. Currently, subjective
evaluation is still the most effective form of evaluating artistic works.
However, subjective evaluation of artistic works will consume a lot of human
and material resources. The popular AI generated content (AIGC) tasks nowadays
have flooded all industries, and music is no exception. While compared to
music produced by humans, AI generated music still sounds mechanical,
monotonous, and lacks aesthetic appeal. Due to the lack of music datasets with
rating annotations, we have to choose traditional aesthetic equations to
objectively measure the beauty of music. In order to improve the quality of AI
music generation and further guide computer music production, synthesis,
recommendation and other tasks, we use Birkhoff’s aesthetic measure to design
a aesthetic model, objectively measuring the aesthetic beauty of music, and
form a recommendation list according to the aesthetic feeling of music.
Experiments show that our objective aesthetic model and recommendation method
are effective.

Computational aesthetics, Music recommendation, Music evaluation, Birkhoff’s
measure

††copyright: acmcopyright††journalyear: 2023††doi: XXXXXXX.XXXXXXX††copyright:
acmlicensed††conference: Proceedings of the 31st ACM International Conference
on Multimedia; October 29–November 3, 2023; Ottawa, ON, Canada.††booktitle:
Proceedings of the 31st ACM International Conference on Multimedia (MM ’23),
October 29–November 3, 2023, Ottawa, ON, Canada††price: 15.00††isbn:
979-8-4007-0108-5/23/10††doi: 10.1145/XXXXXX.XXXXXX††ccs: Applied computing
Sound and music computing

##  1\. Introduction

Computational aesthetic evaluation (Galanter, 2012, 2013) refers to using
computational methods to assess the aesthetic beauty of different types of
art, including paintings, photographs and music. It usually makes qualitative
or quantitative evaluations (Jin et al., 2019) to assess the quality of
artworks generated by AI. It is necessary to consider the aesthetics of
artworks because beauty often make people feel pleasant (Deng et al., 2017).

Figure 1. We construct a music recommendation list based on the results of the
proposed music aesthetic model by adding four aesthetic features into music
recommendation task.

In recent years, recommendation systems have flourished in various fields (Sun
et al., 2019), including music related fields. Nowadays, when faced with such
a massive amount of music resources, music recommendation can help users
quickly find music of interest (Shen et al., 2020). In fact, traditional
recommendation systems often recommend music based on user preferences or the
popularity of the music (Schedl et al., 2015), which may overlook some music
works with higher artistic value.

Generally speaking, the beauty of artistic works, especially the beauty of
music, is highly abstract and subjective. The earliest research on music
aesthetics is proposed by mathematicians (tracing back to Pythagoras) and
attempt to solve it by abstracting it into emotional expression (Meyer, 1956)
or language (Lerdahl and Jackendoff, 1996).

We attempt to propose a method to objectively evaluate the beauty of music.
Fig.1 briefly shows our work.

A musical art work usually needs to go through three steps from creation to
hearing like Fig.2: Firstly, the musician composes and arranges music to make
music score. Secondly, the performer plays music score to get performance.
Finally, rendering the performance into sound according to different
instrument selections (Oore et al., 2020).

However, these music data are not labeled with aesthetic ratings like AVA
(Murray et al., 2012) or AADB (Kong et al., 2016) in the field of image
aesthetic evaluation. So we can’t predict the beauty of music due to the lack
of subjective labeled data. Due to the lack of datasets with subjective rating
labels, we try to objectively measure the beauty of music. We adopt
traditional measure to quantify the beauty of music.

Figure 2. Three steps from creation to hearing for a music art work, and this
paper focuses on the sound stage.

In this paper, Birkhoff’s method (Birkhoff, 2013) was selected to conduct a
study of aesthetic quality assessment of music. The reason why we choose
Birkhoff’s aesthetic measure is that it is not only famous, but also
convenient to formalize, and has a relatively stronger interpretability for
beauty (perhaps the most important reason). Birkhoff formalizes the aesthetic
measure of an object into the quotient between order and complexity:

(1) $M=\frac{O}{C}$

The taste and music creation of users are highly dependent on various factors,
so traditional music recommendation systems often cannot meet users’ needs and
cannot provide good recommendations. Meeting users’ music and entertainment
needs requires considering their contextual (Baltrunas et al., 2011) and good
interactive information (Kulkarni and Rodd, 2020). Therefore, what we focus on
is that our recommendation method should not only make recommendations more
accurate but also recommend more aesthetically pleasing music to users.

The main contributions of our work are as follows:

  * •

We creatively put forward an objective method to measure the beauty of
instrumental music and singing music.

  * •

We propose some basic music features and 4 aesthetic music features to
facilitate the following computational aesthetic evaluation research tasks.

  * •

We apply the music aesthetics model to music recommendation tasks to recommend
more aesthetically pleasing music to users, thereby enhancing their aesthetic
taste.

##  2\. Related Work

###  2.1. Music Generation

Music generation tasks is a critical research field of music information
retrieval (MIR). The music generation task (Ji et al., 2020) mainly includes
three stages: score generation, performance generation and audio generation,
and at present, the main methods of using deep learning to generate music
include CNN (Oord et al., 2016; Yang et al., 2017), RNN (Waite et al., 2016;
Hadjeres et al., 2017; Jeong et al., 2019), VAE (Roberts et al., 2018; Brunner
et al., 2018), GAN (Dong et al., 2018; Huang et al., 2022) and Transformer
(Huang et al., 2018; Shih et al., 2022; Zhang et al., 2022).

The contribution of WaveNet (Oord et al., 2016) in the field of audio
generation is revolutionary, mainly based on its ability to generate high-
quality audio waveforms, which can be used to generate audio and songs for
various instruments. However, models such as WaveNet (Oord et al., 2016),
SampleRNN (Mehri et al., 2016), and NSynth (Engel et al., 2017) are very
complex, require a large number of parameters, a long training time, and a
large number of examples, and lack interpretability. So other models
considered, such as Dieleman et al. (Dieleman et al., 2018) proposed the use
of ADAs to capture long-term correlations in waveforms.

The spectral characteristics of singing are also influenced by F0 motion (Yi
et al., 2019). Previously successful singing synthesizers are based on the
concatenation method (Bonada et al., 2016). Although the systems are state-of-
the-art in terms of sound quality and naturalness, their flexibility is
limited and it is difficult to scale or significantly improve. The statistical
parameter method (Oura et al., 2010) uses a small amount of training data for
model adaptation. But these systems cannot match the existing smooth frequency
and time.

###  2.2. Music Evaluation

Objective evaluation is mainly used to assess the performance of music
generation models. Objective evaluation metrics are designed to standardize
the results of different models, allowing for comparison of their performance.
These metrics are often statistical and can be categorized into four
categories: pitch-related, rhythm-related, harmony-related, and style
transfer-related (Ji et al., 2020), as used in models like MuseGAN (Dong et
al., 2018). In addition, there are specialized metrics that do not directly
relate to music, such as MV2H (McLeod and Steedman, 2018), which evaluates the
performance of music transcription models.

Subjective evaluation is still the best way to assess aesthetics, and its
methods include listening tests and visual analysis. Common listening tests
include the Turing test (Hadjeres et al., 2017; Cífka et al., 2019) and side-
by-side rating (Haque et al., 2018), which provide subjective evaluations of
the music’s quality and degree of innovation, as well as aspects such as
harmony, rhythm, structure, coherence, and overall rating (Dong et al., 2018).

There is currently limited research related to music aesthetics evaluation.
SAAM (Jin et al., 2023a) is a state-of-the-art music aesthetics assessment
model that provides aesthetic ratings for music scores. PAAM (Jin et al.,
2023b) is also a state-of-the-art music aesthetic model for evaluating the
beauty of instrumental performances. Audio Oracle (Dubnov et al., 2011) uses
Information Rate (IR) as an aesthetic measure.

Figure 3. The overview of our aesthetic approach. We align and preprocess the
midi, audio, and mel spectrogram of music, and obtain the values of ten basic
music features by calculating their basic attributes. Then we put the
corresponding basic music features to four logistic regression models to
extract four music aesthetic features. Finally, we take the values of four
music aesthetic features as the input of Birkhoff O/C model, and solve the
model to get the final aesthetic score.

###  2.3. Music Recommendation

Combining deep learning technology with traditional recommendation methods can
effectively improve the effectiveness of music recommendation in recent years.

Van et al. (Van den Oord et al., 2013) utilized the historical listening data
of users and the audio signal data of music to project users and music into a
shared hidden space, enabling them to learn the hidden representations of
users and songs. Jiang et al. (Jiang et al., 2017) proposed an improved
algorithm based on RNN to measure the similarity between different songs. Wang
et al. (Wang and Wang, 2014) proposed a DBN based method that combines the
implicit representations of users and items into a user rating matrix for
model training and improving the performance of music recommendations.
Sachdeva et al. (Sachdeva et al., 2018) proposed a novel attention neural
structure that utilizes the features of these tracks to better understand
users’ short-term preferences.

##  3\. Music Aesthetic Model

###  3.1. Formalization

Based on Birkhoff’s measure, we propose four aesthetic features: harmony,
symmetry, chaos and redundancy. We linearly combine the order measures of
molecules and the complexity measures of denominators by four logistic
regression model. Fig.3 shows our method. Detailed measures explanation will
be described for next. The music aesthetic measure formalization is as
follows:

(2) $Aesthetic\
Measure=\frac{\omega_{1}H+\omega_{2}S+\theta_{1}}{\omega_{3}C+\omega_{4}R+\theta_{2}}$

Where $H$ is harmony, $S$ is symmetry, $C$ is chaos and $R$ is redundancy.
$\omega$ is the weight and $\theta$ is the constant.

###  3.2. Harmony

People feel good because there is “harmony” in music (Clemente et al., 2022).

####  3.2.1. Timbre Harmony

The formalization of timbre harmony can be divided into two aspects: timbre
balance and complementarity.

Timbre balance refers to the relative balance of sound energy in different
frequency bands among instruments.

Let $S_{i}(f)$ denote the sound energy of the $i$-th instrument in frequency
band $f$, and let $F$ be the set of all frequency bands. Then, timbre balance
can be represented as:

(3) $B=\max_{f\in
F}\left(\frac{\max_{i=1}^{n}S_{i}(f)}{\min_{i=1}^{n}S_{i}(f)}\right)$

Where $n$ is the number of instruments, $B$ represents the ratio of the
minimum and maximum sound energy proportions among different frequency bands,
and $B\in[0,1]$.

Timbre complementarity refers to the mutual complementation of sound colors
among instruments.

Let $s_{i}(t)$ denote the sound waveform of the $i$-th instrument at time $t$,
and let $s(t)$ denote the total sound waveform of the ensemble. Then, timbre
complementarity can be represented as:

(4) $C=\frac{1}{T}\int_{0}^{T}\sum_{i=1}^{n}s_{i}(t)s(t)dt$

Where $T$ is the total length of the music, and $C$ represents the timbre
complementarity score among instruments, with $C\in[-1,1]$.

(5) $Timbre\ Harmony=\alpha\cdot B+(1-\alpha)\cdot C$

Timbre harmony is obtained by linearly weighting balance and complementarity.
Where $\alpha$ is the weighting factor.

####  3.2.2. Interval Harmony

The term ”interval” refers to the distance between two notes in music. An
interval of 12 semitones is called an octave. Mathematical and physical
studies have demonstrated that when two sound frequencies have a simple
integer ratio, they produce a more pleasing auditory experience when played
together. To measure the harmony of intervals, We refer to the SAAM (Jin et
al., 2023a) method and propose the following formula.

(6) $Interval\ Harmony=\sum_{i=1}^{12}\alpha_{i}*pir_{i}+\theta_{ih}$

In this formula, $\alpha_{i}$ represents the weight of each interval,
$pir_{i}$ is the ratio of the specific interval to the total number of
intervals, and $\theta_{ih}$ is a constant term.

####  3.2.3. Chord Progression Harmony

In tonal music, harmony progression is a specific harmonic range in chords
connection.

In this study, we adopt the method in SAAM (Jin et al., 2023a), which
calculates the average value of progression tension to obtain a quantitative
chord progress harmony. The formula is as follows:

(7) $\begin{split}Chord\ Progression\
Harmony=\lambda_{1}d_{1}(T_{i},T_{i-1})+\\\
\lambda_{2}d_{2}(T_{i},T_{key})+\lambda_{3}d_{3}(T_{i}-T_{key},T_{f})+\lambda_{4}c(T_{i})+\\\
\lambda_{5}m(T_{i},P)+\lambda_{6}h(T_{i},P)\end{split}$

Here, $T_{i}$ represents the i-th chord of the progression $P$, and $\lambda$
is the weight parameter. Further details on the parameters $c$, $m$, and $h$
can be found in (Navarro-Cáceres et al., 2020).

####  3.2.4. Dynamic Harmony

According to GTTM theory (Lerdahl and Jackendoff, 1996), the metric structure
of a piece of music can be derived from it. The strength of playing each note
can be described as dynamic. The metrical structure of music represents the
strong and weak beats.

We refer to the PAAM (Jin et al., 2023b) approach to quantify the harmony of
dynamics. The formula for dynamic harmony is:

(8) $Dynamic\ Harmony=\frac{\sum_{i=1}^{n}D_{i}\times
M_{i}}{\sqrt{\sum_{i=1}^{n}(D_{i})^{2}}\times\sqrt{\sum_{i=1}^{n}(M_{i})^{2}}}$

Here, $i$ represents the $i$-th note at the beginning of the bar, $n$ is the
total number of aligned notes at the beginning of the bar, $D$ represents the
vector of dynamic values, and $M$ represents the vector of metrical structure
values.

###  3.3. Symmetry

Symmetry is considered to be the key to the perfection of art (Osborne, 1986).

####  3.3.1. Self-Similarity-Fitness

In music, symmetry usually refers to a form of regularity, repetition, or
mirroring in the music. In the music generation field, the structure of a
piece is often regarded as a crucial aspect. Repetitive patterns can be found
in almost all kinds of music. Our aim is to find how the repetitive structure
affects the beauty of music.

Drawing inspiration from the aesthetics of art and images (al Rifaie et al.,
2017), we hypothesize that the beauty of music originates from the symmetry
present in its compositions. Thus, we adopt Müller’s fitness method (Müller et
al., 2013) to gauge the extent of repetition in a musical piece. The fitness
formula is expressed as follows:

(9) $Self\ Similarity\
Fitness=2\cdot\frac{\bar{\sigma}(\alpha)\cdot\bar{\gamma}(\alpha)}{\bar{\sigma}(\alpha)+\bar{\gamma}(\alpha)}$

Where $\bar{\sigma}(\alpha)$ and $\bar{\gamma}(\alpha)$ refer to Müller’s
method (Müller et al., 2013).

###  3.4. Chaos

When people deal with complicated information, they will feel uncomfortable
and uneasy (Bense, 1960).

####  3.4.1. Shannon Entropy

We employ Shannon entropy to quantify the degree of chaos or disorder in the
internal state of a system, using histogram entropy as a specific instance of
this measure. Given a finite set $\Omega$ and a random variable $X$ taking
values in $\Omega$, with each value $x\in\Omega$ having a probability
distribution $p(x)=Pr[X=x]$. What we need to consider is the entropy of the
music attribute histogram. The Shannon entropy $H(X)$ is defined as:

(10) $Shannon\ Entropy=-\sum_{x\in\Omega}p(x)\log(x)$

The Shannon entropy is a widely used method to assess the average uncertainty
of a random variable $X$.

####  3.4.2. Spectral Complexity

Spectral complexity is a method for measuring the complexity of a musical
signal’s spectral content. It is based on the spectral flux of the signal. It
is a measure of how much the spectral content of a signal changes over time.

The spectral complexity of the signal is then defined as the weighted sum of
the spectral flux:

(11) $Spectral\
Complexity=\frac{1}{T}\sum_{t=1}^{T}\sum_{f=1}^{F}|X(f,t)-X(f,t-1)|$

Where $x(t)$ is a signal with $N$ samples, $X(f,t)$ is its short-time Fourier
transform, $F$ is the number of frequency bins, $T$ is the number of analysis
frames.

####  3.4.3. Timbre Variability

Timbre variability is a method used to measure the diversity of timbre in
music. It can calculate the degree of timbre variation in the music.

Formally, let $X={x_{1},x_{2},...,x_{n}}$ represent the $n$ notes in the
music, $f(x_{i})$ represent the basic timbre feature of the $i$th note, and
$Var(x_{i})$ represent the timbre variability of the $i$th note. Then the
Timbre variability of the entire piece of music can be represented as:

(12) $Timbre\ Variability=\frac{\sum_{i=1}^{n}Var(x_{i})\cdot
w_{i}}{\sum_{i=1}^{n}w_{i}}$

Here, $w_{i}$ represents the weight of the $i$th note.

###  3.5. Redundancy

Aesthetically speaking, redundancy makes people feel dull, resulting in
negative emotions (Lorand, 2002).

####  3.5.1. Kolmogorov Complexity

The Kolmogorov complexity $K(s)$ of a string $s$ is defined as the length of
the shortest program that outputs $s$ on a computer. In the context of music,
lossless compression can be used to estimate the redundancy of a piece of
music. This can be expressed using the following formula:

(13) $Kolmogorov\ Complexity=1-\frac{K}{I_{m}}$

Where $K$ is the amount of information after lossless compression of music,
and $I_{m}$ is the original amount of information of music.

####  3.5.2. Autocorrelation Value

The autocorrelation value reflects the degree of redundancy in music.

Given a music signal $x(t)$, the autocorrelation value can be calculated as:

(14) $Autocorrelation\
Value=\frac{1}{N}\sum_{i=0}^{N-1}\frac{\sum_{t=0}^{T-i-1}x(t)x(t+i)}{\sum_{t=0}^{T-1}x^{2}(t)}$

Where $N$ is the maximum time lag for calculating the autocorrelation, $T$ is
the length of the music signal, and $x(t)$ is the amplitude of the signal at
time $t$.

Figure 4. The overview of our recommendation approach. In addition to
traditional positional features and music features, we will also extract good
aesthetic features as important reference standards for sequential music
recommendations. Moreover, we also consider the aesthetic rating score of
music as a label value in the recommendation of sequential music.

##  4\. Aesthetic-aware Recommendation

We use CL4SRec (Xie et al., 2022) as the backbone network of our method, which
is the state-of-art model for sequence recommendation tasks.

###  4.1. Embedding Layer

####  4.1.1. Positional Embedding

We transform positional features into vector representations. We assign a
unique number to each position in the sequence and map each of these to a
d-dimensional vector space to obtain position’s vector representation.

Specifically, we use a matrix $E_{pos}$ of size $L\times d$ to represent the
vector representations of all positions, where $E_{pos_{i}}$ represents the
vector representation for position $i$. We map the position identifier to the
$i$-th row of this matrix, $E_{pos_{i}}$.

Then, we obtain the representation for each position $\mathbf{X}_{pos}$. Then
we will discuss the embedding methods for other features.

####  4.1.2. Other Features Embedding

For the music features and the four aesthetic features, we can use
corresponding embedding layers to convert them into fixed-length vector
representations.

Specifically, we use an embedding matrix $E_{music}$ of size $D_{music}$ to
represent the vector representation of music features (CNN), where $D_{music}$
is the embedding dimension. For each music feature $j$, we represent it as a
vector $\mathbf{v}j\in\mathbb{R}^{D{music}}$, and then map it to a row of the
embedding matrix $E_{music}$, i.e., $E_{music_{j}}=\mathbf{v}_{j}$.

Similarly, we use four embedding matrices $E_{aes}$ of size $D_{aes}$ to
represent the vector representation of aesthetic features.

Then, we obtain the representation for music feature $\mathbf{X}_{music}$ and
aesthetic features $\mathbf{X}{aes}$.

####  4.1.3. Embedding Overview

We add these vector representations with the position embedding vector
representations to form the input matrix $\mathbf{X}$ of the Transformer
model:

(15)
$\mathbf{X}=\mathbf{X}^{(0)}+\mathbf{X}{pos}+\mathbf{X}{music}+\mathbf{X}{aes}$

Where $\mathbf{X}^{(0)}$ represents the vector representation for the original
input sequence, $\mathbf{X}{pos}$ is the position embedding matrix, and
$\mathbf{X}{music}$ and $\mathbf{X}{aes}$ are the embedding matrices for music
and aesthetic features, respectively.

###  4.2. Transformer Layer

####  4.2.1. Multi-Head Self-Attention

For each attention head $h$, we compute the attention weights $\mathbf{A}^{h}$
as follows:

(16)
$\mathbf{A^{h}}=\textbf{softmax}\left(\frac{QW_{i}^{h}(KW_{i}^{h})^{T}}{\sqrt{d_{k}}}\right)$

Where $\mathbf{Q},\mathbf{K},\mathbf{V}\in\mathbb{R}^{n\times d_{k}}$ are the
query, key, and value matrices obtained by linearly projecting the input
sequence $\mathbf{X}$ with learned weight matrices
$\mathbf{W}_{q},\mathbf{W}_{k},\mathbf{W}v\in\mathbb{R}^{d\times d_{k}}$.

The output of the Multi-Head Self-Attention layer is then obtained by
concatenating the $h$ attention heads and passing the result through a linear
layer:

(17) $\textbf{MultiHead(Q, K,
V)}=\text{Concat}(\text{head}_{1},\text{head}_{2},...,\text{head}_{h})\textbf{W}_{o}$

(18)
$\text{head}_{i}=\textbf{Attention}(Q\mathbf{W}_{i}^{q},K\mathbf{W}_{i}^{k},V\mathbf{W}_{i}^{v})$

Where $\mathbf{W}o\in\mathbb{R}^{hd_{v}\times d}$ is a learned weight matrix.

####  4.2.2. Position-wise Feed-Forward Network

The feed-forward neural network consists of two fully connected layers with an
activation function applied after each layer. It can be written as:

(19)
$\mathbf{FFN(x)}=\mathbf{GELU}(\mathbf{x}\mathbf{W}_{1}+\mathbf{b}_{1})\mathbf{W}_{2}+\mathbf{b}_{2}$

(20) $\mathbf{GELU(x)}=x\Phi(x)$

Where $\mathbf{W_{1}}$ and $\mathbf{W_{2}}$ are both learnable weight
matrices, $\mathbf{b}1$ and $\mathbf{b}2$ are the bias terms for the first and
second layers, respectively.

####  4.2.3. Stacking Transformer Layer

The layer consists of $L$ identical layers. Each layer applies a multi-head
self-attention mechanism followed by a position-wise feed-forward network
(FFN) to the input sequence. Additionally, each layer employs residual
connections and layer normalization to stabilize the training process.

The output of the $l$-th layer is given by:

(21)
$\begin{split}\mathbf{X_{l}}=\textbf{LayerNorm}\big{(}\mathbf{X_{l-1}}+\textbf{MultiHead}\big{(}\textbf{LayerNorm}\big{(}\mathbf{X_{l-1}}\big{)}\big{)}\big{)}+\\\
\textbf{FFN}\big{(}\textbf{LayerNorm}\big{(}\mathbf{X_{l-1}}+\textbf{MultiHead}\big{(}\textbf{LayerNorm}\big{(}\mathbf{X_{l-1}}\big{)}\big{)}\big{)}\big{)}\end{split}$

Where $l\in{1,2,...,L}$ indicates the $l$-th layer of the stacking transformer
layer. $\text{LayerNorm}(\cdot)$ represents the layer normalization operation,
$\text{MultiHead}(\cdot)$ represents the multi-head self-attention, and
$\text{FFN}(\cdot)$ represents the position-wise feed-forward network.

###  4.3. Output Layer

For the music sequence recommendation task, the output layer can be formulated
as follows: given the final output $H_{L}$ from the L layers of the
hierarchical exchange information, we mask the item $v_{t}$ at time step $t$
and then use $h_{Lt}$ to predict the masked item $v_{t}$. We apply a two-layer
feed-forward network with GELU activation in between to produce an output
distribution over target items:

(22)
$\mathbf{P(v_{t})}=\textbf{softmax}\left(W_{P}\cdot\text{GELU}\left(h_{Lt}\cdot
E\right)+b_{P}\right)+b_{O}$

Where $\mathbf{W}_{P}$ is the learnable projection matrix, $\mathbf{b}_{P}$
and $\mathbf{b}_{O}$ are bias terms, and $\mathbf{E}\in\mathbb{R}^{|V|\times
d}$ is the embedding matrix for the item set $V$.

###  4.4. Model Learning

We train the model using the randomly masked language modeling (LM) technique,
where we randomly mask portions of song IDs and train the model to predict the
original value of the masked song IDs using the context about non-masked song
IDs in the sequence:

(23) $Input\ layer[\text{s1,s2,s3,s4,s5}]\rightarrow[\text{s1,s2, [mask1],s4,
[mask2]}]$

The cross-entropy function can be represented as follows:

(24) $J=-\frac{1}{N}\sum_{i=1}^{N}\log
P(v_{ti}|v_{1i},...,v_{t-1,i},v_{t+1,i},...,v_{ni})$

Here, $N$ represents the number of training samples, and $v_{ti}$ represents
the masked music item in the $i$-th training sample.

##  5\. Experiments

###  5.1. Datasets & Setup

####  5.1.1. Dataset For Aesthetic Model

The dataset is divided into three types of samples: real audio played or sung
by humans (positive), performance midi rendered audio (medium), and audio
generated by various AI models (negative), 7:3 for training and testing.

Specifically, our positive sample uses MedleyDB (Bittner et al., 2014), our
mid value is audio rendered (by MuseScore3) from POP909’s (Wang et al., 2020)
piano and Lakh MIDI Dataset’s (Raffel et al., 2016) vocal performance midi,
and the negative sample uses Music Transformer (Huang et al., [n. d.]) to
generate performance midi, and then use GanSynth (Engel et al., 2019) to
synthesize it into audio.

AI (negative) | Rendered (medium) | Human (positive)
---|---|---
122 | 140 | 128
Table 1. The number of samples in the aesthetic model.

####  5.1.2. Recommendation Datasets

We select three datasets:

  * •

MSD (Bertin-Mahieux et al., 2011) dataset is a collection of metadata and
feature analysis data for one million songs, including information about
artists, albums, tracks, lyrics, genres, tonality, and more.

  * •

Last. fm-1k (Celma, 2008) dataset contains implicit feedback data from users
on music, including music play records, music tags, user information, and
more.

  * •

Last. fm-360k (Cantador et al., 2011) dataset adds user-artist relationship
and user information on top of the 1k dataset.

We clean the dataset by removing playback events from the session information
that lack user or information, and only keeping session information with
playback events greater than 10. Then we crawl corresponding audio source
files according to playlists, and use Onsets and Frames (Hawthorne et al.,
2017) to get the corresponding MIDI.

Figure 5. The first two rows in the figure show the distribution of ten basic
music features on three sample sets. The last graph in the third row shows the
aesthetic score distribution of our trained model on the three sample sets.
The remaining four graphs in the third row show the distribution of four
aesthetic features on the three sample sets.

####  5.1.3. Evaluation Metrics

For the aesthetic model scoring task, we evaluate the performance of the
classification model using classification accuracy and F1 measure. For
recommended tasks, we use evaluation metrics to assess the ranking list
generated by the models, which include Hit Ratio (HR), Normalized Discounted
Cumulative Gain (NDCG). As we only have one true item for each user, HR@k is
equivalent to Recall@k and proportional to Precision@k. Additionally, we
report HR and NDCG at k = 1, 5, and 10, where a higher value indicates better
performance.

###  5.2. Implementation Details

####  5.2.1. Music Aesthetic Model

We use music21111https://github.com/cuthbertLab/music21 to obtain music
attributes, Musescore3222https://musescore.org/ to render music, and jSymbolic
(McKay et al., 2018), jAudio (McKay et al., 2005), and librosa (McFee et al.,
2015) to extract corresponding music features to complete the calculation of
basic music features. For calculation, we use the aligned midi, audio, and
mel. Regarding the four aesthetic features, we will linearly combine the
corresponding basic music features and use four logistic regression models for
pre training to obtain aesthetic features. Finally, input the value of
aesthetic features into the O/C model, predict the label, select the cross
entropy as the loss function, and use the Adam (Kingma and Ba, 2014) optimizer
for gradient descent. We set the learning rate to 5e-5, and after 1000
iterations, the loss function converges. For more details, please refer to the
implementation of SAAM (Jin et al., 2023a) and PAAM (Jin et al., 2023b).

####  5.2.2. Music Recommendation Model

We use word2vec to process positional features, CNN to process music features,
and for four aesthetic features, we use the original vector. We train the
model using TensorFlow with truncated normal distribution initialization in
the range of [-0.01, 0.01] for all parameters. The optimization is performed
using the Adam (Kingma and Ba, 2014) with a learning rate of 1e-5,
$\beta_{1}=0.9$, $\beta_{2}=0.999$, $\ell_{2}$ weight decay of 0.01, and
linear decay of the learning rate. The gradient is clipped when its $\ell_{2}$
norm exceeded a threshold of 5. To ensure fair comparison, we set the layer
number $L=2$ and head number $h=4$. We select the mask proportion $\rho$
through validation, resulting in $\rho$ = 0.6 for MSD, and $\rho$ = 0.4 for
Last.fm-1k and Last.fm-360k. Model is trained on NVIDIA GeForce GTX 1080Ti GPU
with a batch size of 256.

###  5.3. Results Analysis & Discussion

####  5.3.1. Music Aesthetic Model

The distribution of various indicators and the final aesthetic measure in the
aesthetic model are shown in Fig.5. After training our O/C model, we observe
the intersection of the aesthetic scores distributions (in the last of the
third row in Fig. 5) is very small, indicating the O/C model’s necessity. To
be more specific, TH, IH, and DH have achieved good differentiation effects,
proving that human music audio is significantly superior to rendered audio and
AI generated audio in terms of harmony. As for the symmetry aspect, the audio
generated by AI has a significantly lower repeatability and appears to be
structurally disorganized, while the differentiation between the other two is
not as significant. In terms of Chaos, the differentiation of basic music
features is not particularly significant. In terms of redundancy, it is clear
that human audio is the best, followed by the other two. The accuracy of our
model on the test set is 92.3%, and the f1 measure is 90.5%.

####  5.3.2. Music Recommendation Model

In Table2, we present a summary of the best performance achieved by all models
on four benchmark datasets. As the NDCG@1 results are equivalent to HR@1 in
our experiments, we have omitted them from the table. We have also applied
some other networks (7 baseline models) as backbone networks, namely BPR
(Rendle et al., 2012), CDAE (Wu et al., 2016), FPMC (Rendle et al., 2010),
GRU4Rec (Hidasi et al., 2015), Caser (Tang and Wang, 2018), SASRec (Kang and
McAuley, 2018), Bert4Rec (Sun et al., 2019) and the performance comparison
will be provided in Table2. The experimental results show that the performance
is best when using CL4SRec as the backbone network, CL4SRec is superior in
nearly all metrics. Note that in the results of Table2, we only add aes
features on CL4SRec. The penultimate line is CL4SRec without aesthetic
features, and our performance has slightly improved after integrating
aesthetic features, proving the aesthetic features are meaningful.

Datasets | Metric | BPR | CDAE | FPMC | GRU4Rec | Caser | SASRec | BERT4Rec | CL4SRec | CL4SRec w/ aes
---|---|---|---|---|---|---|---|---|---|---
MSD | HR@1 | 0.0465 | 0.0479 | 0.0474 | 0.0493 | 0.0577 | 0.0703 | 0.0714 | 0.0936 |  0.0927
HR@5 | 0.1182 | 0.1248 | 0.1242 | 0.1319 | 0.1681 | 0.1692 | 0.1828 |  0.2179 | 0.2185
HR@10 | 0.1978 | 0.2014 | 0.2148 | 0.2369 | 0.2654 | 0.2687 | 0.2584 |  0.3071 | 0.3089
NDCG@5 | 0.0941 | 0.0754 | 0.0831 | 0.0768 | 0.1146 | 0.1195 | 0.1282 |  0.1583 | 0.1624
NDCG@10 | 0.1023 | 0.1273 | 0.1252 | 0.1342 | 0.1564 | 0.1532 | 0.1591 | 0.1902 |  0.1895
Last.fm-1k | HR@1 | 0.0443 | 0.0517 | 0.0392 | 0.0374 | 0.0633 | 0.0669 | 0.0704 |  0.0982 | 0.1014
HR@5 | 0.1138 | 0.1532 | 0.1309 | 0.1414 | 0.1795 | 0.1863 | 0.2384 |  0.2709 | 0.2718
HR@10 | 0.2042 | 0.2251 | 0.2389 | 0.2446 | 0.2647 | 0.2986 | 0.3613 |  0.3999 | 0.4026
NDCG@5 | 0.1045 | 0.0985 | 0.1079 | 0.0963 | 0.1408 | 0.1675 | 0.1501 |  0.1821 | 0.1902
NDCG@10 | 0.1152 | 0.1203 | 0.1269 | 0.1291 | 0.1832 | 0.1859 | 0.1952 | 0.2254 |  0.2202
Last.fm-360k | HR@1 | 0.0768 | 0.1248 | 0.1275 | 0.1481 | 0.1535 | 0.1858 | 0.2151 |  0.2854 | 0.2936
HR@5 | 0.2984 | 0.3095 | 0.3730 | 0.4118 | 0.4528 | 0.4890 | 0.5272 |  0.5894 | 0.5931
HR@10 | 0.4109 | 0.4627 | 0.5219 | 0.5809 | 0.5976 | 0.6012 | 0.6467 |  0.6972 | 0.6981
NDCG@5 | 0.1879 | 0.2173 | 0.2418 | 0.3012 | 0.3623 | 0.3471 | 0.3767 | 0.4484 |  0.4313
NDCG@10 | 0.2164 | 0.2579 | 0.2841 | 0.3448 | 0.3687 | 0.3512 | 0.4176 |  0.4807 | 0.4984
Table 2. The performance of various methods on next-item prediction is
compared, with the best scores in each row shown in bold and the second-best
scores underlined. Note that, we only add aes features on CL4SRec, which
achieves best performance.

###  5.4. Subjective Evaluation

We refer to this work (Dong et al., 2022). To confirm the effectiveness of our
aesthetic model and recommendation algorithm, we conduct a subjective
evaluation experiment and invite 15 professional musicians to participate. It
is necessary to confirm the validity of the dataset, that is, whether the
audio generated by AI has the lowest aesthetic value, followed by rendered
audio, and the audio actually performed or sung by humans has the highest
aesthetic value. We select 5 songs from each of the above three sample sets
and ask volunteers to rate their aesthetic value. The results are shown in
Table3:

AI (negative) | Rendered (medium) | Human (positive)
---|---|---
2.32 ± 0.27 | 3.25 ± 0.32 | 4.18 ± 0.16
Table 3. Results of listening test 1. It presents the Mean Opinion Scores
(MOS) along with a 95% confidence interval.

After proving the validity of our dataset, then we select five different songs
corresponding to the model rating of five levels and ask volunteers to
subjectively rate them, and this will determine whether our aesthetic model
can have good musical aesthetic differentiation ability, shown as Table4:

Low | Medium Low | Medium | Medium High | High
---|---|---|---|---
1.46 ± 0.17 | 2.65 ± 0.32 | 3.24 ± 0.27 | 3.82 ± 0.31 | 4.32 ± 0.26
Table 4. Results of listening test 2. It presents the Mean Opinion Scores
(MOS) along with a 95% confidence interval.

The subjective experimental results demonstrate that our aesthetic model is
effective and can measure the beauty of music. As for the recommendation
algorithm, we also briefly ask volunteers to conduct rating experiments, and
93.3% of volunteers believe that our recommendation algorithm can recommend
more aesthetically pleasing music, proving its effectiveness.

###  5.5. Ablation Study

We conduct ablation study to remove harmony, symmetry, chaos, and redundancy
to train four different models. We compare them with our full model. The
following Table5 shows the results:

Our Full Model |  0.674
---|---
-w/o harmony |  1.028
-w/o symmetry |  0.723
-w/o chaos |  0.785
-w/o redundancy |  0.836
Table 5. Comparisons of the final MSE between the ablation models and the
ground truths, in log scales.

Experiments have shown that our full model has the best performance, with
harmony being the most important aesthetic feature. Regarding the performance
ablation of the model, after removing aes features, there is no significant
performance decline .

##  6\. Conclusion

In this article, we propose an O/C based music aesthetic evaluation model and
propose some music features. In addition, we also integrate our music
aesthetic features for music aesthetic recommendation task. Nevertheless,
there are still shortcomings in this work, such as the lack of consideration
of ”innovation”. We hope that this article may help improve the AIGC music
generation task.

###### Acknowledgements.

We thank the ACs, reviewers and the AI Music Group of BIGAI. This work is
partially supported by the Natural Science Foundation of China
(62072014&62106118), the Fundamental Research Funds for the Central
Universities (3282023014), the Project of Philosophy and Social Science
Research, Ministry of Education of China (20YJC760115), and the Science and
Technology Project of the State Archives Administrator (2022-X-069).

##  Appendix A Pesudo Features

Due to the fact that our music basic features include both midi and audio
features, the data to be evaluated in the model is often audio data, and the
data to be evaluated cannot to be aligned. So, in order to solve the problem
of only audio input data, we have designed a method to create pesudo features
to replace aesthetic features to complete the following aesthetic calculation
tasks (In Fig6).

Figure 6. From aesthetic features to pesudo features, the actual features used
for aesthetic scoring and recommendation are actually pesudo features.

Firstly, we train the model using aligned midi and audio data, and calculate
the values of four aesthetic features using the algorithms in the aesthetic
model.

Next, we use the a2m algorithm to transcribe audio data with missing aligned
midi data, resulting in pesudo midi data.

Then, we treat pesudo midi and audio data as aligned data to extract pesudo
features and evaluate a pesudo rating.

Finally, we use 4 aesthetic features and 4 pesudo features as loss, while
using both aesthetic and pesudo ratings as loss to make the value of pesudo
data as close as possible to the value of GT data.

##  Appendix B Extra Experimental Results

Below are experimental data and results of the pesudo features task:

Metric | Aligned M & A | Pesudo M & A | A2M M & A
---|---|---|---
Accuracy | 92.3% | 91.7% | 84.2%
F1-Measure | 90.5% | 90.1% | 79.8%
Table 6. The table compares the performance of three types of data in aesthetic rating tasks. From left to right, they are: Aligned Midi and Audio data, Pesudo Midi and Audio data, and A2M (method in previous) Midi and Audio data. Dataset | HR@1 | HR@5 | HR@10 | NDCG@5 | NDCG@10
---|---|---|---|---|---
MSD | 0.0931 | 0.2188 | 0.3076 | 0.1633 | 0.1892
Last.fm-1k | 0.1017 | 0.2699 | 0.4034 | 0.1911 | 0.2197
Last.fm-360k | 0.2928 | 0.5925 | 0.6980 | 0.4351 | 0.4988
Table 7. The table data shows the performance of using CL4SRec as the backbone
network to fuse pesudo aesthetic features in recommendation tasks.

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# Anomaly and Brownian fluid particle in Navier-Stokes turbulence

Timo Aukusti Laine<EMAIL_ADDRESS>
###### Abstract

We investigate the Navier-Stokes turbulence driven by a stochastic random
Gaussian force. Using a field-theoretic approach, we uncover an anomaly that
brings hidden structure to the theory. The anomaly is generated by a non-self-
adjoint operator of the Jacobian and it follows the symmetries of the
stochastic Navier-Stokes equation. We calculate the anomaly and demonstrate
that by forcing the anomaly to vanish, the velocity field is constrained and a
monopole-type object with a constant charge is formed. When the viscosity is
zero, the anomaly can be interpreted as the Brownian damping coefficient of a
random fluid particle. We provide the Brownian particle equation and its
solution in the presence of a pump and viscosity. Our results suggest that the
anomaly is an inherent feature of stochastic turbulence and must be taken into
account in all stochastic turbulence calculations. This constitutes an
additional law for the original set of stochastic Navier-Stokes equations.

##  I Introduction

Navier-Stokes turbulence is a fascinating and complex phenomenon that
represents one of the most challenging and elusive problems in classical
physics. It is a branch of fluid dynamics that seeks to clarify the chaotic
and turbulent motion of fluids, which is characterized by rapid and irregular
swirling of fluid particles. These phenomena are typically described by the
Navier-Stokes equations, partial differential equations that govern the
conservation of momentum and mass in fluid flow. Although these equations have
helped to understand turbulence, due to their nonlinearity, they are still an
active area of research in flow dynamics.

Navier-Stokes turbulence within field theory represents an intersection of
flow dynamics and theoretical physics. This approach extends the traditional
understanding of the Navier-Stokes equations to field theory and brings a new
perspective to the behavior of turbulent flows. It allows us to dive into the
complex interaction of various physical quantities in a turbulent flow, from
velocity and pressure fields to vortices and energy distribution. This unique
perspective is invaluable in understanding the statistical properties of
turbulence that often elude traditional approaches. The application of field
theory to turbulence brings a new perspective, where fluid properties are
treated as fields and an attempt is made to understand their behavior through
quantum field theory, statistical mechanics or other field theoretical
approaches, Refs. frisch, -ramond, .

In this article we will explore the stochastic Navier-Stokes turbulence and
extend the results of Burgers turbulence to cover the Navier-Stokes equation,
Ref. laine, .

##  II Navier-Stokes turbulence

In this section, we derive the generating functional of correlation functions
based on the stochastic Navier-Stokes equation. This is a basis function that
can be used to calculate various types of correlations in turbulent motion,
Ref. zinn, .

Notations

We explore the following stochastic Navier-Stokes equations

$\displaystyle{\rm A)}$ $\displaystyle\frac{\partial{\vec{U}}}{\partial
t}+({\vec{U}}\cdot\nabla){\vec{U}}=\frac{1}{\rho}\nabla
p+\nu\nabla^{2}{\vec{U}}+{\vec{f}},$ (1) $\displaystyle{\rm B)}$
$\displaystyle\nabla\cdot\vec{U}=0,$ (2)

where

$\vec{U}={\vec{U}}(t,\vec{x})=u_{1}\hat{i}+u_{2}\hat{j}+u_{3}\hat{k}$ (3)

is the velocity field, $\rho$ is the mass density, $p$ is the pressure, $\nu$
is the viscosity, and $\vec{f}=\vec{f}(t,\vec{x})$ is the Gaussian random
force satisfying the condition

$\langle
f_{i}(t,{\vec{x}})f_{j}(t^{\prime},{\vec{y}})\rangle=\kappa({\vec{x}}-{\vec{y}})\delta_{ij}\delta(t-t^{\prime}).$
(4)

Function $\kappa(\vec{x}-\vec{y})$ defines the spatial correlation of the
random forces. While the density is constant, the fluid is assumed
incompressible (2). We introduce the notation

$x_{\mu}\hskip 14.22636pt\mu=0,1,2,3\hskip 5.69046pt{\rm with}\hskip
5.69046ptx_{0}=t,\hskip 8.5359pt(x_{1},x_{2},x_{3})=(x_{i})=\vec{x},$ (5)

i.e.

$x_{\mu}=(x_{0},x_{i})=(t,\vec{x}),\hskip 8.5359pti=1,2,3.$ (6)

We write equation (1) as

$\partial_{t}u_{i}+(u_{j}\partial_{j})u_{i}=\partial_{i}\tilde{p}+\nu\partial_{j}\partial_{j}u_{i}+f_{i},$
(7)

where $\tilde{p}=p/\rho$, $\partial_{j}=\partial/\partial x_{j}$ and
$i=1,2,3$. We have also used a shortened notation

${\vec{U}}\cdot\nabla=\sum_{j=1}^{3}u_{j}\partial_{j}=u_{j}\partial_{j},$ (8)

where repeated indices are summed expect when otherwise indicated.

Generating functional

The generating functional of correlation functions is defines as, Ref. zinn, ,

$\langle F[\lambda]\rangle=\int DfDuF[\lambda]\exp(-S[f,u]).$ (9)

The $\langle F[\lambda]\rangle$ can be written with the help of
$\delta$-functions (when $i=1$),

$\displaystyle\int
Df_{1}Du_{1}\delta(\partial_{t}u_{1}+(u_{j}\partial_{j})u_{1}-\partial_{1}\tilde{p}-\nu\partial_{j}\partial_{j}u_{1}-f_{1})J[u]\exp\Bigl{(}-\frac{1}{2}\int
dtd^{3}xd^{3}yf_{1}(t,\vec{x})\kappa^{-1}(\vec{x}-\vec{y})f_{1}(t,\vec{y})\Bigr{)}$
(10) $\displaystyle=\int Du_{1}D\mu_{1}J[u]\exp\Bigl{(}-\frac{1}{2}\int
dtd^{3}xd^{3}y\mu_{1}(t,\vec{x})\kappa(\vec{x}-\vec{y})\mu_{1}(t,\vec{y})+i\int
d^{4}x\mu_{1}(\partial_{t}u_{1}+(u_{j}\partial_{j})u_{1}-\partial_{1}\tilde{p}-\nu\partial_{j}\partial_{j}u_{1})\Bigr{)}.$

Here, $J[u]$ is the Jacobian or determinant of the function inside the delta
function. Since $\vec{U}$ is a vector, we could write 3 similar equations for
each axis $1,2,3$. Alternatively, we introduce a matrix or metric $h^{ij}$
which is zero except for $h^{11}=\hat{i}$ , $h^{22}=\hat{j}$ , and
$h^{33}=\hat{k}$ and has the property

$\mu^{i}=h^{ij}\mu_{j}.$ (11)

Also, for example,

$\mu^{i}f_{i}=\mu_{1}f_{1}\hat{i}+\mu_{2}f_{2}\hat{j}+\mu_{3}f_{3}\hat{k}.$
(12)

Then we may write the generating functional as

$\langle F[\lambda]\rangle=\int DuD\mu F[\lambda]J[u]\exp(-S[u,\mu]),$ (13)

where the action $S[u(t,\vec{x}),\mu(t,\vec{x})]$ is

$S[u,\mu]=\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p}-\nu\partial_{j}\partial_{j}u_{i}).$
(14)

$S$ is a kind of vector, but it can be understood as consisting of 3 different
equations, all of which are described by one equation.

Jacobian

We determine the Jacobian $J[u]$. It is not necessarily obvious what the
functional derivative of a vector is. Usually a functional derivative is
applied to the scalar field. That’s why we use the definition

$\vec{U}=u\vec{e},$ (15)

where $\vec{e}=e_{1}\hat{i}+e_{2}\hat{j}+e_{3}\hat{k}$ and write the Jacobian
of the stochastic Navier-Stokes equation as

$J[u]=\det\biggl{|}\frac{\delta\vec{f}}{\delta
u}\biggr{|}=\frac{\delta}{\delta u}\Bigl{(}\frac{\partial u}{\partial
t}+u(e_{j}\partial_{j})u-\nu\partial_{j}\partial_{j}u\Bigr{)}\vec{e}.$ (16)

We have dropped the pressure term because it does not depend on $u$. The
Jacobian is also a vector function consisting of a determinant and a unit
vector $\vec{e}$. The functional derivative can now be calculated and the
determinant becomes

$J[u]=\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|\vec{e}.$
(17)

The value of the determinant is the same for all three directions. By using
anticommuting functions, $\Psi=\Psi(t,\vec{x})$ and
$\tilde{\Psi}=\tilde{\Psi}(t,\vec{x})$, Refs. zinn, -ramond, , the determinant
can be written as

$J[u]=\int D\bar{\Psi}D\Psi\exp(-S_{A}),$ (18)

where the action of the determinant is

$S_{A}=-\vec{e}\int
d^{4}x\bar{\Psi}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\Psi.$
(19)

The action is also a vector with 3 components.

Full action

We collect the results. Action (14) together with the Jacobian (19) form the
complete action,

$\displaystyle S_{full}[u,\mu,\Psi,\tilde{\Psi}]=$
$\displaystyle\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p}-\nu\partial_{j}\partial_{j}u_{i})$
(20) $\displaystyle-\vec{e}\int
d^{4}x\bar{\Psi}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\Psi.$

The action (20) has all the information of the stochastic Navier-Stokes
equation in one equation.

##  III Effective action when viscosity is zero

In this section we calculate the determinant or Jacobian (17). First, it
should be noted that the determinant depends on the velocity field $u$

$\frac{\delta}{\delta u}S_{A}=\vec{e}e_{j}\partial_{j}\bar{\Psi}\Psi\neq 0,$
(21)

so it cannot be ignored. The problem is that the operator in the determinant
is non-self-adjoint, and if we want to express the action without the Jacobi
fields, i.e. using only $u$, we need to calculate the square of the
determinant. This is analogous to chiral field theories. Therefore we first
need to define the ”right-handed” and ”left-handed” actions or determinants
after which the square can be calculated.

Chiral determinants

We observe that

$\int
d^{4}x\bar{\Psi}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\Psi=\int
d^{4}x(-\partial_{t}-u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\bar{\Psi}\Psi.$
(22)

We get the right-handed determinant, which is

$S_{R}=-\vec{e}\int
d^{4}x\bar{\Psi}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\Psi,$
(23)

and the left-handed determinant is

$S_{L}=\vec{e}\int
d^{4}x(-\partial_{t}-u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\bar{\Psi}\Psi=-\vec{e}\int
d^{4}x\Psi(-\partial_{t}-u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})\bar{\Psi}.$
(24)

An additional ”$-$” sign comes from the integral,

$\int D\bar{\Psi}D\Psi\exp^{S[\bar{\Psi},\Psi]}=-\int D\Psi
D\bar{\Psi}\exp^{-S[\Psi,\bar{\Psi}]}=\int D\Psi
D\bar{\Psi}\exp^{S[\Psi,\bar{\Psi}]}.$ (25)

The square of the chiral determinant is

$\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|=\Bigl{[}\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|\det|-\partial_{t}-u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|\Bigr{]}^{1/2}.$
(26)

The causality needs to be preserved when calculating the determinant.

Square of the determinant

First we set the viscosity to zero. The challenge of calculating the square of
the determinant (26) is that the usual methods of two-dimensional chiral
methods are not applicable. They use complex values and vectors, but now all
is real. Alternatively, we require that $S_{R}=S_{L}$ and use the definition
of the determinant, which is the determinant is the product of its
eigenvalues. Therefore, we consider the following eigenvalue equations

$\begin{cases}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j}))A=\lambda
A,\\\ (-\partial_{t}-u(e_{j}\partial_{j}))A=\lambda A,\end{cases}$ (27)

where $\lambda=\lambda(t,\vec{x},u)$ is an eigenvalue and $A=A(t,\vec{x},u)$
is the corresponding eigenfunction. For consistency, both determinants must
produce the same eigenvalue functions. Putting the equations (27) together
gives

$\lambda=\frac{1}{2}(e_{j}\partial_{j})u(t,\vec{x}).$ (28)

By analogy with chiral theories, the result is already the ”square root”. The
determinant with the Jacobi fields is then

$S_{D}(\nu=0)=-\vec{e}\int
d^{4}x\bar{\Psi}\frac{1}{2}(e_{j}\partial_{j})u(t,\vec{x})\Psi.$ (29)

One can use symmetries to verify that the result is correct, Ref. laine, . An
alternative way to derive the anomaly is to note that

$\frac{\delta}{\delta u}S_{D}=\frac{\delta}{\delta
u}S_{R}+\frac{\delta}{\delta u}S_{L}.$ (30)

As the definition of the determinant states, the value of the determinant is
then

$\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})|=\frac{1}{2}(e_{j}\partial_{j})u(t,\vec{x}).$
(31)

The determinant cannot be zero, because then the generating functional (13) is
zero and we have no physics (or some type of regularization is needed to treat
the divergence). We write the determinant as a path-integral. By using the
identity

${\rm detM}=\exp{{\rm Tr}[{\rm lnM}}],$ (32)

where M is a matrix, we may write

${\rm detM}=\exp(-S_{D}),$ (33)

and

$S_{D}(\nu=0)=-\vec{e}\int
d^{4}x\ln(\frac{1}{2}e_{j}\partial_{j}u(t,\vec{x})).$ (34)

This is a path-integral representation of the square root of the determinant.
The anomaly is due to the second term of the stochastic Navier-Stokes equation
(1). This is consistent with the observation that

$\vec{\mu}\cdot(\vec{U}\cdot\nabla)\vec{U}$ (35)

can also be understood as the interaction Lagrangian of a charged particle
with a Dirac monopole or a magnetic point vortex, Ref. jackiw, .

Constraint

In path-integrals, local symmetry transformations are particularly interesting
because they describe physical phenomena typically observed in nature. We have
shown that in general the determinant of the stochastic Navier-Stokes equation
depends on the velocity field (31) and it cannot be neglected. However, one
can consider situations where the anomaly is forced to vanish, i.e. the
determinant is required to be a number which cause constraints on the field
$u$. The requirement of the determinant to be constant gives the condition

$\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})|=\frac{1}{2}e_{j}\partial_{j}u(t,\vec{x})={\rm
constant}.$ (36)

We call this ”fixing the gauge”, i.e. we have chosen the determinant to be a
constant number. We could select some other gauge as well. Later we will show
that if we want to examine the symmetries of the stochastic Navier-Stokes
turbulence equation, this gauge is the only correct choice.

From (36) we derive the constraint condition

$\frac{\delta}{\delta u}{\rm det}=0,$ (37)

or

$(e_{j}\partial_{j})\delta u(t,\vec{x})=0.$ (38)

These are the conditions we expect from the anomaly. The variation of the
function (34) is

$\delta S_{D}=-2\vec{e}\int
d^{4}x\frac{1}{e_{i}\partial_{i}u(t,\vec{x})}e_{j}\partial_{j}\delta
u(t,\vec{x}).$ (39)

This integral is zero if

$\displaystyle\delta u={\rm constant},\hskip 14.22636pt{\rm or}$ (40)
$\displaystyle e_{j}\partial_{j}\delta u=0,\hskip 14.22636pt{\rm or}$ (41)
$\displaystyle e_{j}\partial_{j}e_{i}\partial_{i}u=0.$ (42)

The equations show the constraint correctly.

Zero eigenvalue

Zero eigenvalues are especially interesting because they typically enable some
additional physical property or observable. In the stochastic Navier-Stokes
turbulence there is a zero eigenvalue in the determinant. We have the
eigenvalue equation (27)

$(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j}))A(t,\vec{x},u)=\lambda
A(t,\vec{x},u)=\frac{1}{2}(e_{j}\partial_{j})u(t,\vec{x})A(t,\vec{x},u).$ (43)

From (43) we get the zero eigenvalue equation

$(\partial_{t}+\frac{1}{2}e_{j}\partial_{j}u+ue_{j}\partial_{j})A(t,\vec{x},u)=0\cdot
A(t,\vec{x},u)=0.$ (44)

We note that the constraint imposes no restrictions on $t$. Therefore we have

$\displaystyle\partial_{t}A(t,\vec{x},u)=0,\hskip 14.22636pt{\rm and}$ (45)
$\displaystyle
ue_{j}\partial_{j}A(t,\vec{x},u)=-\frac{1}{2}e_{j}\partial_{j}uA(t,\vec{x},u).$
(46)

We will use these values later when calculating the viscosity of the anomaly.

Effective action

We collect the results when the viscosity is zero. The effective action is

$S_{eff}[\nu=0]=\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p})-\vec{e}\int
d^{4}x\bar{\Psi}\frac{1}{2}e_{j}\partial_{j}u\Psi,$ (47)

or

$S_{eff}[\nu=0]=\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p})-\vec{e}\int
d^{4}x\ln(\frac{1}{2}e_{j}\partial_{j}u(t,x)).$ (48)

The constraint

$e_{j}\partial_{j}\delta u(t,\vec{x})=0$ (49)

is a condition for which the anomaly vanishes. Equation (47) is the same as
the equation (20), but in (47) the determinant is a square root, which can
then be expressed in terms of the field $u$, and this is the equation (48).

##  IV Effective action when viscosity is non-zero

Next, we consider the situation where the viscosity is non-zero. Again, we
need to calculate the square root of the determinant. The action with the
Jacobi fields is

$\displaystyle S_{eff}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p}-\nu\partial_{j}\partial_{j}u_{i})$
(50) $\displaystyle-\vec{e}\int
d^{4}x\bar{\Psi}(\frac{1}{2}e_{j}\partial_{j}u-\nu\partial_{j}\partial_{j})\Psi.$

We calculate the determinant. We use the following set of eigenvalue equations

$\begin{cases}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})A^{\prime}=\lambda^{\prime}A^{\prime},\\\
(-\partial_{t}-u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j})A^{\prime}=\lambda^{\prime}A^{\prime},\end{cases}$
(51)

where $\lambda^{\prime}=\lambda^{\prime}(t,\vec{x},u)$ is the eigenvalue and
$A^{\prime}=A^{\prime}(t,\vec{x},u)$ is the corresponding eigenfunction. This
gives the result

$\lambda^{\prime}=\frac{1}{2}e_{j}\partial_{j}u(t,\vec{x})-\nu\frac{\partial_{j}\partial_{j}A^{\prime}}{A^{\prime}}=\lambda-\nu\frac{\partial_{j}\partial_{j}A^{\prime}}{A^{\prime}},$
(52)

where $\lambda$ is the eigenvalue of the non-viscosity equations. We add (52)
back into the equation (51). This gives

$\begin{cases}(\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j}))A^{\prime}=\lambda
A^{\prime},\\\ (-\partial_{t}-u(e_{j}\partial_{j}))A^{\prime}=\lambda
A^{\prime},\end{cases}$ (53)

from which it follows that $A^{\prime}=A$, and $A$ is the eigenvector of the
non-viscosity equations. This means that even in the presence of viscosity,
the viscosity does not couple to or change the eigenvector. We have then

$\lambda^{\prime}=\frac{1}{2}e_{j}\partial_{j}u-\nu\frac{\partial_{j}\partial_{j}A}{A}.$
(54)

The condition (46) can be solved with respect to $A$

$A(t,\vec{x},u)=Au(t,\vec{x})^{-1/2}.$ (55)

The eigenvalue $\lambda^{\prime}$ can be calculated

$\lambda^{\prime}=\frac{1}{2}e_{j}\partial_{j}u-\nu
F[u]=\frac{1}{2}e_{j}\partial_{j}u-\nu\Bigl{(}\frac{3}{4}\frac{(\partial_{j}u)^{2}}{u^{2}}-\frac{\partial_{j}\partial_{j}u}{2u}\Bigr{)}.$
(56)

The Jacobian is

$J[u]=\exp\Bigl{(}\vec{e}\int
d^{4}x\ln(\frac{1}{2}e_{j}\partial_{j}u-\nu\Bigl{(}\frac{3}{4}\frac{(\partial_{j}u)^{2}}{u^{2}}-\frac{\partial_{j}\partial_{j}u}{2u}\Bigr{)})\Bigr{)},$
(57)

and the effective action then becomes

$\displaystyle S_{eff}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dtd^{4}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}-\partial_{i}\tilde{p}-\nu\partial_{j}\partial_{j}u_{i})$
(58) $\displaystyle-\vec{e}\int
d^{4}x\ln(\frac{1}{2}e_{j}\partial_{j}u-\nu\Bigl{(}\frac{3}{4}\frac{(\partial_{j}u)^{2}}{u^{2}}-\frac{\partial_{j}\partial_{j}u}{2u}\Bigr{)})\
.$

When the velocity field is complex differentiable in an open set, it can be
shown by the power series that

$F[u]=\frac{3}{4}\frac{(\partial_{j}u)^{2}}{u^{2}}-\frac{\partial_{j}\partial_{j}u}{2u}>0.$
(59)

The viscosity becomes part of the constraint.

##  V Brownian particle

When the viscosity is zero, the action integral can be written without the
anomaly term, i.e. by forcing the anomaly to vanish. We reformulate the action
of the stochastic Navier-Stokes equation so that it does not contain the
anomaly.

Notations

We define an object $L$,

$L=L[u]=e_{j}\partial_{j}u.$ (60)

$L$ has the property

$L={\rm constant}\neq 0,$ (61)

and it does not change as $u$ varies

$\frac{\delta}{\delta u}L=0,\hskip 14.22636pt{\rm or}\hskip 14.22636pt\delta
L=0.$ (62)

Equations (61) and (62) are the anomaly constraint conditions. Additionally,
$L$ has symmetry properties

$\tilde{L}[\tilde{t},\tilde{\vec{x}},\tilde{u}]=f_{1}(t,\vec{x},u)L[f_{2}(t,\vec{x},u)],$
(63)

where $f_{1}$ and $f_{2}$ are some functions. These are shown in the next
section.

Fluid particle

We may now write

$(\vec{U}\cdot\nabla)\vec{U}=ue_{j}\partial_{j}u\vec{e}=uL\vec{e}=L\vec{U}.$
(64)

Note that the equation (64) can be interpreted as an eigenvalue equation
$\vec{U}\cdot\nabla$ with eigenvalue $L$ and eigenvector $\vec{U}$. The
stochastic Navier-Stokes equation (1) becomes

$\frac{\partial{\vec{U}}}{\partial t}+L\vec{U}=\frac{1}{\rho}\nabla
p+\nu\nabla^{2}{\vec{U}}+{\vec{f}}.$ (65)

When setting $p=0$ and $\nu=0$ in (65) we get

$\frac{\partial{\vec{U}}}{\partial t}+L\vec{U}={\vec{f}}.$ (66)

This equation describes Brownian motion, the seemingly random motion of a
particle in a fluid caused by collisions with molecules in the fluid, Refs.
coffey, -zinn, . $L$ is the damping coefficient. The general solution of the
equation of motion is

$\vec{U}(t)=\vec{U}(0)e^{-Lt}+\int_{0}^{t}dt^{\prime}\vec{f}(t^{\prime})e^{-(t-t^{\prime})}.$
(67)

We can add the pressure term to the equation (66)

$\frac{\partial{\vec{U}}}{\partial t}+L\vec{U}-\frac{1}{\rho}\nabla
p={\vec{f}},$ (68)

and the solution is

$\vec{U}(t)=\vec{U}(0)e^{-Lt}+\int_{0}^{t}dt^{\prime}[\vec{f}(t^{\prime})e^{-(t-t^{\prime})}+\frac{1}{\rho}\nabla
p].$ (69)

Brownian action

Next, we look at the path-integral representation. The path-integral (50)
without the anomaly is

$S_{eff}[\nu=0]=\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+Lu_{i}-\partial_{i}\tilde{p}).$ (70)

We consider the determinant and use the result (56)

$\displaystyle\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}L[u]-\nu F[u]$ (71) $\displaystyle=$
$\displaystyle{\rm constant}-\nu F[U]$ (72) $\displaystyle=$ $\displaystyle
1-\nu F[U].$ (73)

Here we have added a constant to the right-hand side of the determinant, which
together with $L[u]/2$ forms the constant $1$. This addition applies only when
$L[U]$ is constant and examining a change in $u$ in the determinant where

$\frac{\delta}{\delta
u}\det|\partial_{t}+(e_{j}\partial_{j})u+u(e_{j}\partial_{j})-\nu\partial_{j}\partial_{j}|=-\nu\frac{\delta}{\delta
u}F[u].$ (74)

The action of the determinant is then

$\displaystyle S_{D}$ $\displaystyle=$ $\displaystyle-\vec{e}\int
d^{4}x\ln(1-\nu F[u])$ (75) $\displaystyle\approx$ $\displaystyle-\vec{e}\int
d^{4}x\Bigl{(}(-\nu F[u])-\frac{1}{2}(-\nu F[u])^{2}+...\Bigr{)}.$ (76)

In (76) we have used the expansion of the logarithmic function. The path-
integral can be written in the form

$\displaystyle S_{eff}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+L[u]u_{i}-\partial_{i}\tilde{p}-\nu\partial_{j}\partial_{j}u_{i})$
(78) $\displaystyle-\vec{e}\int d^{4}x\ln(\frac{1}{2}L[u]-\nu F[u])$
$\displaystyle\approx$ $\displaystyle\frac{1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,{\vec{x}})\kappa({\vec{x}}-{\vec{y}})\mu_{i}(t,{\vec{y}})-i\int
d^{4}x\mu^{i}(\partial_{t}u_{i}+L[u]u_{i}-\partial_{i}\tilde{p})$
$\displaystyle+\nu\int
d^{4}x(i\mu^{i}\partial_{j}\partial_{j}u_{i}+\vec{e}F[u]).$

Here we have a fluid particle in random Brownian motion which is affected by
the pump (first term) and the viscosity (last term).

##  VI Symmetry properties of $L$

In this section, we investigate how the damping coefficient $L$ changes under
the Navier-Stokes symmetries. The stochastic Navier-Stokes equation (1),
$f=0$, is invariant with the following symmetries, Ref. cantwell, .

a. | Space-translations | $\vec{U}(t,\vec{x})=\vec{U}(t,\vec{x}+\vec{a})\hskip 14.22636pt\vec{a}\in\mathbb{R}^{3}$
---|---|---
b. | Time-translations | $\vec{U}(t,\vec{x})=\vec{U}(t+\tau,\vec{x})\hskip 14.22636pt\tau\in\mathbb{R}$
c. | Space-reflections (parity) | $P\vec{U}(t,\vec{x})=-\vec{U}(t,-\vec{x})$
d. | Galilean transformations | $\vec{U}(t,\vec{x})=\vec{U}(t,\vec{x}+\vec{a}t)-\vec{a}\hskip 14.22636pt\vec{a}\in\mathbb{R}^{3}$
e. | Space-rotations | $\vec{U}(t,\vec{x})=R^{T}\vec{U}(t,R\vec{x})\hskip 14.22636ptR\in SO(3)$
f. | Scale invariance in space ($\nu=0$) | $\vec{U}(t,\vec{x})=\lambda^{h}\vec{U}(t,\lambda^{h}\vec{x})\hskip 14.22636pt\lambda\in\mathbb{R}^{+}$
g. | Scale invariance in time ($\nu=0$) | $\vec{U}(t,\vec{x})=\lambda^{-h}\vec{U}(\lambda^{h}t,\vec{x})\hskip 14.22636pt\lambda\in\mathbb{R}^{+}$

Next, we consider how these symmetries affect $L$.

a. Space-translations

The generator of the space translations is

$\delta u_{i}=a_{j}\partial_{j}u_{i},$ (79)

where $a_{j}$ is constant. The variation of the damping coefficient is then

$\delta L=e_{j}\partial_{j}\delta
u_{i}e_{i}=e_{k}\partial_{k}(a_{j}\partial_{j}u_{i})e_{i}=a_{j}\partial_{j}L=0,$
(80)

and the $L$ is

$\tilde{L}(t,\vec{x})=L(t,\vec{x}+\vec{a})={\rm constant}.$ (81)

b. Time-translations

In a same way, the generator of the time-translation is

$\delta u_{i}=a\partial_{t}u_{i},$ (82)

and the $L$ is

$\tilde{L}(t,\vec{x})=L(t+a,\vec{x})={\rm constant}.$ (83)

c. Parity

Parity cannot be represented by a sequence of infinitesimal transformations.
It’s easy to see though

$\tilde{L}(t,\vec{x})=L(t,-\vec{x})={\rm constant}.$ (84)

d. Galilean symmetry

The infinitesimal change is

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
a_{j}t\partial_{j}u_{i}-a_{i},$ (85) $\displaystyle\delta\mu_{i}$
$\displaystyle=$ $\displaystyle a_{j}t\partial_{j}\mu_{i},$ (86)
$\displaystyle\delta\Psi_{i}$ $\displaystyle=$ $\displaystyle
a_{j}t\partial_{j}\Psi_{i},$ (87) $\displaystyle\delta\bar{\Psi}_{i}$
$\displaystyle=$ $\displaystyle a_{j}t\partial_{j}\bar{\Psi}_{i},$ (88)

and the damping coefficient gives

$\delta L=e_{j}\partial_{j}\delta u_{i}e_{i}=at\partial_{t}L_{i}e_{i}=0.$ (89)

This corresponds to

$\displaystyle\tilde{L}$ $\displaystyle=$ $\displaystyle
L(t,\vec{x}+\vec{a}t)={\rm constant},$ (90) $\displaystyle\tilde{\vec{u}}$
$\displaystyle=$ $\displaystyle\vec{u}(t,\vec{x}+\vec{a}t)-\vec{a}.$ (91)

e. Space-rotations

The field variations of the space-rotations are

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
a(q_{kl}x_{l}\partial_{k}u_{i}-q_{ik}u_{k}),$ (92)
$\displaystyle\delta\mu_{i}$ $\displaystyle=$ $\displaystyle
a(q_{kl}x_{l}\partial_{k}\mu_{i}+q_{ki}\mu_{k}),$ (93)
$\displaystyle\delta\Psi_{i}$ $\displaystyle=$ $\displaystyle
a(q_{kl}x_{l}\partial_{k}\Psi_{i}-q_{ik}\Psi_{k}),$ (94)
$\displaystyle\delta\bar{\Psi}_{i}$ $\displaystyle=$ $\displaystyle
a(q_{kl}x_{l}\partial_{k}\bar{\Psi}_{i}+q_{ki}\bar{\Psi}_{k}),$ (95)

where $q_{ij}$ is the rotation in $SO(3)$. The change in $L$ is

$\delta L=(R\cdot\nabla)L=0.$ (96)

This means that the damping coefficient is invariant under the rotational
symmetry,

$\tilde{L}(t,\vec{x})=L(t,R\vec{x})={\rm constant}.$ (97)

f. Scale invariance in space ($\nu=0$)

The transformation rules are

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
a(x_{j}\partial_{j}u_{i}-u_{i}),$ (98) $\displaystyle\delta\mu_{i}$
$\displaystyle=$ $\displaystyle a(x_{j}\partial_{j}\mu_{i}+2\mu_{i}),$ (99)
$\displaystyle\delta\tilde{p}$ $\displaystyle=$ $\displaystyle
a(x_{j}\partial_{j}\tilde{p}-2\tilde{p}),$ (100) $\displaystyle\delta\Psi_{i}$
$\displaystyle=$ $\displaystyle
a(x_{j}\partial_{j}\Psi_{i}+\frac{1}{2}\Psi_{i}),$ (101)
$\displaystyle\delta\bar{\Psi}_{i}$ $\displaystyle=$ $\displaystyle
a(x_{j}\partial_{j}\bar{\Psi}_{i}+\frac{1}{2}\bar{\Psi}_{i}).$ (102)

The change in $L$ is

$\tilde{L}(t,\vec{x})=L(t,\lambda^{h}\vec{x})={\rm constant}.$ (103)

g. Scale invariance in time ($\nu=0$)

The transformation rules are

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
a(t\partial_{t}u_{i}+u_{i}),$ (104) $\displaystyle\delta\mu_{i}$
$\displaystyle=$ $\displaystyle a(t\partial_{t}\mu_{i}-\mu_{i}),$ (105)
$\displaystyle\delta\tilde{p}$ $\displaystyle=$ $\displaystyle
a(t\partial_{t}\tilde{p}+2\tilde{p}),$ (106) $\displaystyle\delta\Psi_{i}$
$\displaystyle=$ $\displaystyle at\partial_{t}\Psi_{i},$ (107)
$\displaystyle\delta\bar{\Psi}_{i}$ $\displaystyle=$ $\displaystyle
at\partial_{t}\bar{\Psi}_{i}.$ (108)

The change in $L$ is

$\tilde{L}(t,\vec{x})=\lambda^{-h}L(\lambda^{h}t,\vec{x})={\rm constant}.$
(109)

We have shown that the damping coefficient obeys all the same symmetries that
the stochastic Navier-Stokes equation has. In all transformations, it is
constant, as required by the constraint. The symmetry transformations do not
change the coefficient expect in the scale invariance in time. This is a
feature with implications.

Symmetries of the determinant actions

In this section, we show that $L[u]$ must be constant due to the symmetries of
the stochastic Navier-Stokes equation. First, suppose that $L[u]$ is not
constant.

We start by applying symmetry transformations to the anomaly of the action,
equation (29),

$S_{D1}(\nu=0)=-\vec{e}\int
d^{4}x\bar{\Psi}\frac{1}{2}(e_{j}\partial_{j})u(t,\vec{x})\Psi.$ (110)

As can be verified, using the variations of the fields $u$, $\Psi$ and
$\bar{\Psi}$, the action (110) is invariant under all local symmetry
transformations a.-g.. In other words, for these local symmetries

$\delta S_{D1}(\nu=0)_{\rm a.-g.}=0,$ (111)

and the result is independent of whether $L[u]$ is constant. We then write
(110) as

$S_{D1}(\nu=0)=-\frac{1}{2}\vec{e}\int d^{4}x\bar{\Psi}L[u]\Psi.$ (112)

The variation of (112) is

$\displaystyle\delta S_{D1}(\nu=0)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\vec{e}\int
d^{4}x(\delta\bar{\Psi}L[u]\Psi+\bar{\Psi}\delta
L[u]\Psi+\bar{\Psi}L[u]\delta\Psi)$ (113) $\displaystyle=$
$\displaystyle-\frac{1}{2}\vec{e}\int d^{4}x(\bar{\Psi}\delta
L[u]\Psi+L[u](\delta\bar{\Psi}\Psi+\bar{\Psi}\delta\Psi)).$ (114)

We need and we have an additional condition. Now looking at the eigenvalue
equation (27), we find that

$\Psi=\bar{\Psi},$ (115)

and because the $\Psi$ and $\bar{\Psi}$ are anticommuting fields, the result
is

$\delta S_{D1}(\nu=0)=-\frac{1}{2}\vec{e}\int d^{4}x\bar{\Psi}\delta
L[u]\Psi.$ (116)

Variation (116) is zero only if $\delta L[u]=0$ or $L[u]$ is constant.

We derive the same result in a different way. We have also another formula for
the anomalous action, equation (34),

$S_{D2}(\nu=0)=-\vec{e}\int
d^{4}x\ln(\frac{1}{2}e_{j}\partial_{j}u(t,\vec{x})).$ (117)

The Jacobi fields are missing from this action which enabled the vanishing
action in field variations, (110) and (111). However, for consistency reasons,
for the same local symmetries a.-g., the variation of the action (117) must
also vanish under the same symmetry transformations, i.e.

$\delta S_{D2}(\nu=0)_{\rm a.-g.}=0.$ (118)

Therefore, the constraint condition

$L[u]={\rm constant},$ (119)

is the only valid choice of gauge that the stochastic Navier-Stokes turbulence
equation can have when examining the symmetries of this equation. Otherwise,
the symmetry properties of the anomalous actions (110) and (117) are
different.

In summary, we have shown in two different ways that $L[u]$ must be constant.

##  VII Vortices and monopoles

A Vortex is a stable time-independent solution to a set of classical field
equations that has finite energy in two spatial dimensions; it is a two-
dimensional soliton. In three spatial dimensions, a vortex becomes a string, a
classical solution with finite energy per unit length, Ref. preskill, .

In this section, we examine $L$ for a Brownian fluid particle solution, (67),

$\vec{U}(t,\vec{x})=\vec{U}(0,\vec{x})e^{-Lt}+\int_{0}^{t}dt^{\prime}\vec{f}(t^{\prime},\vec{x})e^{-(t-t^{\prime})}.$
(120)

The constraint $L$ becomes

$L\vec{e}=e_{j}\partial_{j}u\vec{e}=\int_{0}^{t}dt^{\prime}e_{j}\partial_{j}\vec{f}(t^{\prime},\vec{x})e^{-(t-t^{\prime})}={\rm
constant}\cdot\vec{e}.$ (121)

We have assumed that the first term of (120) can be neglected in large values
of time. Taking a derivative with respect to time we have

$e_{j}\partial_{j}f_{i}(t,\vec{x})=0.$ (122)

We assume a pulse which has a Gaussian shape

$f_{i}\sim\exp(-a_{j}x_{j}^{2}),$ (123)

where $a_{i}$ is constant. Equation (122) gives

$\sum_{j=1}^{3}a_{j}e_{j}x_{j}=0.$ (124)

The condition (124) create a constraint on the shape of the soliton. In two
dimensions, the vortex has a Gaussian shape and it is rotationally symmetric
in $u$. In three spatial dimensions, a vortex has a shape

$f_{i}\sim\exp(-a_{1}x_{1}^{2}-a_{2}x_{2}^{2}-[(-a_{1}e_{1}x_{1}-a_{2}e_{2}x_{2})/(a_{3}e_{3})]^{2}),$
(125)

and this is a string.

There are two things to note. First, the stochastic force $f$ is the source of
the anomaly and the vortex. Without a force there are no vortices. And
conversely, when a stochastic force is present, there is an anomaly or vortex
in the theory. Second, as considered earlier in this article,

$\vec{\mu}\cdot(\vec{U}\cdot\nabla)\vec{U}$ (126)

indeed describes the interaction term between the field and the vortex.

##  VIII Brownian particle in scaling symmetry

As a more concrete example, we consider the following local time
reparameterization, $\beta=\beta(t)$, $a$ and $b$ are constants,

$\displaystyle\tilde{t}$ $\displaystyle=$ $\displaystyle\beta(t)^{a},$ (127)
$\displaystyle\tilde{\vec{x}}$ $\displaystyle=$
$\displaystyle\vec{x}\beta^{\prime}(t)^{b},$ (128)
$\displaystyle\tilde{u}_{i}$ $\displaystyle=$ $\displaystyle
u_{i}\beta^{\prime}(t)^{a-b},$ (129) $\displaystyle\tilde{\mu}_{i}$
$\displaystyle=$ $\displaystyle\mu_{i}\beta^{\prime}(t)^{2b-a},$ (130)
$\displaystyle\tilde{\tilde{p}}$ $\displaystyle=$
$\displaystyle\tilde{p}\beta^{\prime}(t)^{2a-2b}.$ (131)

This combines the scaling of time and space symmetries. The transformation
corresponds to the following field variations

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
a\beta\partial_{t}u_{i}+b\beta^{\prime}x_{j}\partial_{j}u_{i}+(a-b)\beta^{\prime}u_{i},$
(132) $\displaystyle\delta\mu_{i}$ $\displaystyle=$ $\displaystyle
a\beta\partial_{t}\mu_{i}+b\beta^{\prime}x_{j}\partial_{j}\mu_{i}+(2b-a)\beta^{\prime}\mu_{i},$
(133) $\displaystyle\delta\tilde{p}$ $\displaystyle=$ $\displaystyle
a\beta\partial_{t}\tilde{p}+b\beta^{\prime}x_{j}\partial_{j}\tilde{p}+(2a-2b)\beta^{\prime}\tilde{p}.$
(134)

The change in the action is

$\displaystyle\frac{\delta S_{eff}}{b}$ $\displaystyle=$
$\displaystyle\frac{3h-1}{2}\int
dtd^{3}xd^{3}y\beta^{\prime}\mu^{i}(t,\vec{x})\kappa(\vec{x}-\vec{y})\mu_{i}(t,\vec{y})-i\int
dx^{4}\beta^{\prime\prime}\mu^{i}(x_{j}\partial_{j}u_{i}-hu_{i})$ (135)
$\displaystyle+i(h+1)\nu\int
dx^{4}\beta^{\prime}\mu^{i}\partial_{j}\partial_{j}u_{i}-\vec{e}\frac{1}{b}\delta\int
dx^{4}\Bigl{(}\frac{1}{2}L[u]-\nu F[u]\Bigr{)},$

where $h=(b-a)/b$ The second term vanishes either if

$\displaystyle{\rm a)}$ $\displaystyle\beta^{\prime\prime}=0,\hskip
56.9055pt{\rm or}$ (136) $\displaystyle{\rm b)}$ $\displaystyle
x_{j}\partial_{j}u_{i}-hu_{i}=C/\beta^{\prime\prime},$ (137)

where $C$ is constant. In case b) the integral becomes zero based on
conservation of motion of the center of mass.

Brownian particle

We set $\beta=t$. Then the change is

$\tilde{u}_{i}=\lambda^{a-b}u_{i}(\lambda^{a}t,\lambda^{b}\vec{x}),$ (138)

and

$\displaystyle\delta u_{i}$ $\displaystyle=$ $\displaystyle
at\partial_{t}u_{i}+bx_{j}\partial_{j}u_{i}+(a-b)u_{i},$ (139)
$\displaystyle\delta\mu_{i}$ $\displaystyle=$ $\displaystyle
at\partial_{t}\mu_{i}+bx_{j}\partial_{j}\mu_{i}+(2b-a)\mu_{i},$ (140)
$\displaystyle\delta\tilde{p}$ $\displaystyle=$ $\displaystyle
at\partial_{t}\tilde{p}+bx_{j}\partial_{j}\tilde{p}+(2a-2b)\tilde{p},$ (141)
$\displaystyle\delta\tilde{L}$ $\displaystyle=$ $\displaystyle
at\partial_{t}L+bx_{j}\partial_{j}L+aL=0.$ (142)

This is a Brownian fluid particle in the presence of the stochastic pump and
the viscosity

$\vec{U}(t)=\vec{U}(0)e^{-L[u]t}+\int_{0}^{t}dt^{\prime}[\vec{g}(t^{\prime})e^{-(t-t^{\prime})}+\frac{1}{\rho}\nabla
p],$ (143)

and the change of the damping coefficient is

$\tilde{L}=\lambda^{a}L(\lambda^{a}t,\lambda^{b}\vec{x}).$ (144)

The force $\vec{g}$ is obtained as an equation of motion from the equation
(14), $\nu=0$,

$\frac{\delta
S}{\delta\mu}=\vec{g}(t)-(\partial_{t}\vec{U}+(\vec{U}\cdot\nabla)\vec{U}-\nabla\tilde{p})=0,$
(145)

and

$\vec{g}(t)=\frac{1}{2}\int
d^{3}xd^{3}y\Bigl{(}\vec{\mu}(t,\vec{x})\kappa(\vec{x}-\vec{y})+\kappa(\vec{x}-\vec{y})\vec{\mu}(t,\vec{y})\Bigr{)}.$
(146)

We have also used the Wick rotation, $\vec{\mu}\to i\vec{\mu}$. Based on (144)
the solution (143) scales in time but not in space. The symmetry is visible
when the pump and the viscosity are in balance

$\frac{\delta S_{eff}}{b}\approx\frac{3h-1}{2}\int
dtd^{3}xd^{3}y\mu^{i}(t,\vec{x})\kappa(\vec{x}-\vec{y})\mu_{i}(t,\vec{y})+\nu\int
d^{4}x(i(h+1)\mu^{i}\partial_{j}\partial_{j}u_{i}+\vec{e}\frac{1}{b}\delta
F[u])=0.$ (147)

We have used the representation of the Brownian random motion (78). The pump
term injects the energy and the viscosity dissipates it.

##  IX Conclusions

Navier-Stokes turbulence is a chaotic and turbulent flow of a fluid
characterized by rapid and irregular movement of fluid particles. On the other
hand, Brownian motion is the random movement of particles suspended in a
liquid, which is caused by their collision with fast-moving atoms or molecules
in the surrounding medium. Random Brownian motion in Navier-Stokes turbulence
represents a complex interaction between two separate phenomena in fluid
dynamics and statistical physics. The connection between Brownian motion and
Navier-Stokes turbulence arises when looking at small particles or tracer
particles suspended in a turbulent fluid.

In this article, we have shown that there is an anomaly in the Navier-Stokes
turbulence driven by the Gaussian random force. The anomaly must somehow be
handled in stochastic turbulence calculations. If no choice is made, an
inherent feature of the anomaly is lost. Therefore, our result suggests a
third law for the original Navier-Stokes equations (1)-(2)

$\displaystyle{\rm A)}$ $\displaystyle\frac{\partial{\vec{U}}}{\partial
t}+({\vec{U}}\cdot\nabla){\vec{U}}=\frac{1}{\rho}\nabla
p+\nu\nabla^{2}{\vec{U}}+{\vec{f}},$ (148) $\displaystyle{\rm B)}$
$\displaystyle\nabla\cdot\vec{U}=0,$ (149) $\displaystyle{\rm C)}$
$\displaystyle{\rm Rule\hskip 2.84526pthow\hskip 2.84526ptto\hskip
2.84526pttreat\hskip 2.84526ptthe\hskip 2.84526ptanomaly\hskip
2.84526pt(=fixing\hskip 2.84526ptthe\hskip 2.84526ptgauge).}$ (150)

We need a rule, Eq. (150), that defines how to fix the ”gauge” for the
velocity field. In this article we selected the gauge that adds a constant
constraint

$\displaystyle{\rm A)}$ $\displaystyle\frac{\partial{\vec{U}}}{\partial
t}+({\vec{U}}\cdot\nabla){\vec{U}}=\frac{1}{\rho}\nabla
p+\nu\nabla^{2}{\vec{U}}+{\vec{f}},$ (151) $\displaystyle{\rm B)}$
$\displaystyle\nabla\cdot\vec{U}=0,$ (152) $\displaystyle{\rm C)}$
$\displaystyle(\vec{e}\cdot\nabla)\vec{U}={\rm constant}.$ (153)

This choice of gauge, Eq. (153), forces the anomaly to be a number and removes
the anomalous term from the theory. Instead, a monopoly-type object is formed,
whose ”charge” is ”constant”. If we want to examine the local symmetries of
the stochastic Navier-Stokes equation, this gauge is the only correct choice.
The gauge (153) is consistent with the Brownian damping coefficient of a
random fluid particle. The difference to the traditional Brownian factor is
that in this gauge selection the coefficient depends on the velocity field and
it also follows all the symmetries of the Navier-Stokes equation. Another
selection of the gauge is also possible, and the choice may depend on the
physical problem under investigation.

## References

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