Particularly for a probability density function belonging to the exponential family, from Eqs. (2) and (3) the covariant coefficients and metric tensor are written as [9, 10].

$$\Gamma\_{ijk}^{(a)}(\theta) = -\frac{(1-a)}{2} \partial\_i \partial\_j \partial\_k \varphi(\theta),\tag{5}$$

$$\mathcal{g}\_{\vec{\eta}}(\theta) = \frac{\partial^2 \boldsymbol{\mu}(\theta)}{\partial \theta\_i \partial \theta\_j}. \tag{6}$$

These coefficients, for *α* ∈ ℝ, determine a one-parameter family of affine connections, and each element of this family is called an *α*-connection. An affine connection allows one to compare vectors in nearby tangent spaces [9]. Moreover, the covariant coefficients satisfy the following relation:

$$
\Gamma\_{ij}^{k(a)} = \mathbf{g}^{km} \Gamma\_{ijm}^{(a)},\tag{7}
$$

where *gkm* � � is the inverse matrix of the metric and *gij* � � and Γ*k*ð Þ *α ij* are the contravariant coefficients.

In the case *α* ¼ 0, the coefficients reduce to the Levi-Civita's connection:

$$
\Gamma^{(0)}\_{ijk} = \frac{1}{2} \left( \partial\_i \mathbf{g}\_{jk} + \partial\_j \mathbf{g}\_{ik} - \partial\_k \mathbf{g}\_{ij} \right). \tag{8}
$$

From Eq. (3) the curvature tensor for an *α*-connection is written as

$$R^{(a)}\_{ijkm} = \left(\partial\_i \Gamma^{s(a)}\_{jk} - \partial\_j \Gamma^{s(a)}\_{ik}\right) \mathbf{g}\_{im} + \left(\Gamma^{(a)}\_{im} \Gamma^{r(a)}\_{jk} - \Gamma^{(a)}\_{jrm} \Gamma^{r(a)}\_{ik}\right). \tag{9}$$

Since Γð Þ *α ijk* ð Þ*θ* vanishes identically for *α* ¼ 1, any exponential family of PDF constitutes an uncurved space when the *α* ¼ 1 connection is used. The one connection is therefore called the exponential connection as we mentioned above [11]. The relationships (Eqs. (5) and (6)) lead to the simplification of the geometric construction associated with the probability density function (Eq. (4)).

For systems in thermodynamic equilibrium, the parameters *θi* may include inverse temperature, chemical potential, pressure, magnetic field, and so on. Also, *ψ θ*ð Þ represents the thermodynamic potential of the system [8]. It is worth noting that, if we consider the equilibrium density functions and *α* ¼ 0, this formalism reproduces the geometric structure found by Ruppeiner for the space of equilibrium states. In the Ruppeiner formalism, the metric is constructed using the theory of fluctuations, and the geometric elements are obtained as the second derivatives of the corresponding thermodynamic potential [2]. In this paper we have chosen the approach of statistical manifold because it allows us to address nonequilibrium problems.

In other words, it is the analysis of the dynamic behavior of the system and the

In the previous context, this chapter will focus on the analysis of the optimal evolution. In particular we consider the Uhlenbeck-Ornstein (UO) nonequilibrium process described by the probability density function (PDF) solution of the Fokker-Planck Equation [6]. From this probability density function, we build a twodimensional metric-affine manifold in the coordinates ð Þ *μ*, *σ* , where *μ* is the mean and *σ* the standard deviation. In this coordinates and for the connections *α* ¼ 0 and

However, due to simplicity in geometric construction, we are interested in studying the behavior of the system in coordinates ð Þ *θ*1, *θ*2 , where *θ*1 ¼ 1*=* 2*σ*2 ð Þ and *θ*2 ¼ �*μ=σ*2. In these coordinates the probability density function belongs to the exponential family, and using this formal expression and analogously with the equilibrium probability density function, we build a macroscopic potential *ψ θ*ð Þ 1, *θ*2 for the UO process. From the geometry constructed using the *ψ θ*ð Þ 1, *θ*2 potential, we show that for *α* ¼ 3 and *α* ¼ 2, the system evolves on a geodesic curve. In particular for *α* ¼ 3, we show that the manifold is flat and for the steady state the macroscopic potential and entropy have the same functional dependence. Thinking the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy

In the second section of this chapter, we summarize the most relevant aspects of the theory of the statistical manifold. The geometric development associated with the fundamental solution of the Fokker-Planck equation of UO process is found in the third section. The fourth section is devoted to the construction of the potential. In the fifth section, we analyze the geometric relationship between macroscopic potential and entropy. In the sixth section, we present our conclusions and perspectives.

In this section we briefly review the information of geometrical theory [4] that is used to analyze geometrically a family of probability density functions (PDF) and its application to thermodynamics. Let *p x*ð Þ , *θ* be a PDF described by a random variable *x* and parameters *θ* ¼ ð Þ *θ*1, *θ*2, … , *θm* that characterize a system. The set of PDFs

becomes an *m*-dimensional statistical manifold having *θi* coordinates. According

In the statistical manifold *M*, we can introduce a natural derivative of the vector

1 � *α*

where *l x*ð Þ¼ , *θ* ln ½ � *p x*ð Þ , *θ* and *E*½ �*:* means the expectation operation with respect to *p x*ð Þ , *θ* . The last expression is obtained by the use of the normalization condition *E ∂i* ½ �¼ *l x*ð Þ , *θ* 0. This metric tensor is the Fisher information matrix in

to the information geometrical theory, we can make a metric tensor *gij*

field *B* toward the tangent vector *A*, denoted by ∇*α*

*ijk* ð Þ¼ *θ E ∂i∂ jl x*ð Þþ , *θ*

*M* ¼ *p x*ð Þ , *θ* : *θ* ∈ Ω ⊂ ℝ*m* f g (1)

*AB*. It is obtained through the

*∂kl x*ð Þ , *θ*

*:* (3)

*gij*ð Þ¼ *θ E ∂il x*ð Þ , *θ ∂ jl x*ð �¼� , *θ*Þ *E ∂i∂ jl x*ð Þ , *θ* , (2)

2 *∂il*ð*x*, *θ*Þ*∂ jl*ð*x*, *θ*Þ

study of the conditions that are optimal.

*Advances on Tensor Analysis and Their Applications*

*α* ¼ �1, the system evolves on a geodesic curve [7].

describes the steady state of an optimal evolution.

**2. Elements of statistical manifold**

information theory.

**74**

covariant coefficients [7]:

Γð Þ *α*

#### **3. Fokker-Planck equation and macroscopic potential**

A diffusion process can be thought of as a process of Uhlenbeck-Ornstein. The Uhlenbeck-Ornstein process is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The probability density function *P x*ð Þ , *t* of the Uhlenbeck-Ornstein process satisfies the Fokker-Planck equation [6]:

$$\frac{\partial P}{\partial t} = \frac{1}{\tau} \frac{\partial}{\partial \mathbf{x}} \left[ (\mathbf{x} - \boldsymbol{\varkappa}\_0) P \right] + D \frac{\partial^2 P}{\partial \mathbf{x}^2}. \tag{10}$$

By analogy with the probability density functions for systems in equilibrium, we

1 *θ*3 1 þ 3*θ*2 2 2*θ*4 1 � �, Γð Þ *α*

211 ¼ ð Þ *α* � 1 2

212 ¼ � ð Þ *α* � 1 4*θ*2 1

Using Eqs. (18) and (19), we calculate the curvature tensor component in the

From Eq. (19) we observe that there are two *α* values for which the manifold is

In a manifold with a connection, we can generalize the straight line of Euclidean geometry. The generalized straight line is called geodesic, and it is defined by the characteristic that its tangent vector does not change its direction. It satisfies the

> *dθ j du* ¼ � *d*2

with an arbitrary parameter *u*. The special parameter *s* is called the affine parameter. We note that if *u* is a linear transformation of *s*, the right-hand side vanishes. Let us now prove that the curve defined by Eq. (15) of the UO process is a geodesic. We choose the Newtonian time *t* for the arbitrary parameter *u*. From Eq. (20) for the study of geodetic curves, we consider that the temporal dependence

> 1 2*Dτ* 1 � *e*�2*t=τ* ð Þ,

> > *x*0 1 � *e*�*t=τ* � � *Dτ* 1 � *e*�2*t=τ* ð Þ*:*

*dt*2 ¼ cothð Þ *t=τ* csch2

ð Þ *t=*ð Þ 2*τ* csch3

ð Þ t*=τ* 2*Dτ*2 (22)

ð Þ t*=τ Dτ*3 *:* (23)

*θ*1ðÞ¼ *t*

8 >>>><

>>>>:

*θ*2ðÞ¼� *t*

ð Þ *t=τ* 4*Dτ*2 , *d*2

*θ*2

The tangent vector coordinate *dθi=dt* and the acceleration coordinate

*θ*1

*dt*2 ¼ 2*x*0 sinh 4

222 ¼ 0

(18)

*:* (19)

*du* (20)

(21)

*θ*2 2*θ*3 1

ð Þ *α* � 3 ð Þ *α* � 1 8*θ*3 1

> *u=ds*2 ð Þ *du=ds* 2

*dθi*

will call the relationship (Eq. (17)) "nonequilibrium potential" [8].

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

111 ¼ � ð Þ *α* � 1 2

Γð Þ *α*

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

Γð Þ *α* 112 ¼ Γð Þ *α*

Γð Þ *α* 122 ¼ Γð Þ *α*

*R*ð Þ *α*

*d*2 *θi du*2 þ Γ*i jk dθk du*

of the coordinates ð Þ *θ*1, *θ*2 is given by

*dθ*1

*dt* ¼ � *x*0sech2

*dt* ¼ � cosh 2

ð Þ *t=*ð Þ 2*τ* 4*Dτ*2 , *d*2

coordinates ð Þ *θ*1, *θ*2

Eq. (11)

*d*2

**77**

*θi=dt*2 are

and

*dθ*2

Using the potential (Eq. (17)) we calculate the coefficients (Eq. (5))

121 ¼ Γð Þ *α*

221 ¼ Γð Þ *α*

1212ð Þ¼� *θ*1, *θ*2

flat, *α* ¼ 3 and *α* ¼ 1. The case *α* ¼ 1 is a direct consequence of Eq. (5).

The fundamental solution of this linear parabolic partial differential equation, and the initial condition consisting of a unit point mass at location *y*, is:

$$P(\mathbf{x},t) = \sqrt{\frac{1}{2\pi D\tau(1 - e^{-2t/\tau})}} \exp\left\{-\frac{\mathbf{1}}{2D\tau} \left[\frac{\left(\mathbf{x} - \mathbf{x}\_0 - (\mathbf{y} - \mathbf{x}\_0)e^{-t/\tau}\right)^2}{\mathbf{1} - e^{-2t/\tau}}\right] \right\} \tag{11}$$

which is the Gaussian density function with mean

$$
\mu = \varkappa\_0 + (\mathcal{y} - \varkappa\_0)e^{-t/\tau} \tag{12}
$$

and variance

$$
\sigma^2 = D\tau \left( 1 - e^{-2t/\tau} \right) \tag{13}
$$

where *x*0 represents the average length of the displacement, *D* is the diffusion coefficient, and *τ* is a characteristic time. Without loss of generality, in the rest of the work, we consider *y* ¼ 0.

Considering Eqs. (12) and (13), we think the function (Eq. (11)) as a probability density function dependent on two parameters *μ* and *σ*, formally:

$$p(\mathbf{x}, \mu, \sigma) = \frac{\mathbf{1}}{\sqrt{2\pi\sigma^2}} e^{-(\mathbf{x}-\mu)^2/2\sigma^2}. \tag{14}$$

Considering these parameters as the coordinates ð Þ *μ*, *σ* of the manifold *M* and taking into account Eqs. (2) and (4), it can be seen that for *α* ¼ 0 and *α* ¼ �1, the system evolves on a geodesic curve [7].

Inspired by the simplicity of relations (Eqs. (5) and (6)), we use an alternative description of the UO process through the coordinates:

$$f(\theta\_1, \theta\_2) = \left(\frac{1}{2\sigma^2}, -\frac{\mu}{\sigma^2}\right). \tag{15}$$

In these coordinates the probability density function (Eq. (14)) belongs to the exponential family and is written as

$$p(\mathbf{x}, \theta\_1, \theta\_2) = \exp\left[-\theta\_1 \mathbf{x}^2 - \theta\_2 \mathbf{x} - \varphi(\theta\_1, \theta\_2)\right] \tag{16}$$

where

$$
\psi(\theta\_1, \theta\_2) = \frac{\theta\_2^2}{4\theta\_1} - \ln\left[\sqrt{\frac{\theta\_1}{\pi}}\right].\tag{17}
$$

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

By analogy with the probability density functions for systems in equilibrium, we will call the relationship (Eq. (17)) "nonequilibrium potential" [8].

Using the potential (Eq. (17)) we calculate the coefficients (Eq. (5))

$$\begin{aligned} \Gamma\_{111}^{(a)} &= -\frac{(a-1)}{2} \left( \frac{1}{\theta\_1^3} + \frac{3\theta\_2^2}{2\theta\_1^4} \right), & \Gamma\_{222}^{(a)} &= \mathbf{0} \\\\ \Gamma\_{12}^{(a)} &= \Gamma\_{121}^{(a)} = \Gamma\_{211}^{(a)} = \frac{(a-1)}{2} \frac{\theta\_2}{2\theta\_1^3} \\\\ \Gamma\_{122}^{(a)} &= \Gamma\_{221}^{(a)} = \Gamma\_{212}^{(a)} = -\frac{(a-1)}{4\theta\_1^2} \end{aligned} \tag{18}$$

Using Eqs. (18) and (19), we calculate the curvature tensor component in the coordinates ð Þ *θ*1, *θ*2

$$R\_{1212}^{(a)}(\\\theta\_1,\\\\\theta\_2) = -\frac{(a-\mathfrak{Z})(a-\mathfrak{1})}{8\theta\_1^3}.\tag{19}$$

From Eq. (19) we observe that there are two *α* values for which the manifold is flat, *α* ¼ 3 and *α* ¼ 1. The case *α* ¼ 1 is a direct consequence of Eq. (5).

In a manifold with a connection, we can generalize the straight line of Euclidean geometry. The generalized straight line is called geodesic, and it is defined by the characteristic that its tangent vector does not change its direction. It satisfies the Eq. (11)

$$\frac{d^2\theta\_i}{du^2} + \Gamma^i\_{jk}\frac{d\theta\_k}{du}\frac{d\theta\_j}{du} = -\frac{d^2u/ds^2}{(du/ds)^2}\frac{d\theta\_i}{du} \tag{20}$$

with an arbitrary parameter *u*. The special parameter *s* is called the affine parameter. We note that if *u* is a linear transformation of *s*, the right-hand side vanishes. Let us now prove that the curve defined by Eq. (15) of the UO process is a geodesic. We choose the Newtonian time *t* for the arbitrary parameter *u*. From Eq. (20) for the study of geodetic curves, we consider that the temporal dependence of the coordinates ð Þ *θ*1, *θ*2 is given by

$$\begin{cases} \theta\_1(t) = \frac{1}{2D\tau(1 - e^{-2t/\tau})}, \\\\ \theta\_2(t) = -\frac{\varkappa\_0 \left(1 - e^{-t/\tau}\right)}{D\tau(1 - e^{-2t/\tau})}. \end{cases} \tag{21}$$

The tangent vector coordinate *dθi=dt* and the acceleration coordinate *d*2 *θi=dt*2 are

$$\frac{d\theta\_1}{dt} = -\frac{\cosh^2(t/\tau)}{4D\tau^2}, \quad \frac{d^2\theta\_1}{dt^2} = \frac{\coth(t/\tau)\text{csch}^2(t/\tau)}{2D\tau^2} \tag{22}$$

and

$$\frac{d\theta\_2}{dt} = -\frac{\varkappa\_0 \text{sech}^2(t/(2\tau))}{4D\tau^2}, \quad \frac{d^2\theta\_2}{dt^2} = \frac{2\varkappa\_0 \sinh^4(t/(2\tau))\text{csch}^3(t/\tau)}{D\tau^3}.\tag{23}$$

**3. Fokker-Planck equation and macroscopic potential**

*∂P ∂t* ¼ 1 *τ ∂*

which is the Gaussian density function with mean

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2*πDτ* 1 � *e*�2*t=τ* ð Þ

*Advances on Tensor Analysis and Their Applications*

Fokker-Planck equation [6]:

s

the work, we consider *y* ¼ 0.

system evolves on a geodesic curve [7].

exponential family and is written as

where

**76**

description of the UO process through the coordinates:

*P x*ð Þ¼ , *t*

and variance

A diffusion process can be thought of as a process of Uhlenbeck-Ornstein. The Uhlenbeck-Ornstein process is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The probability density function *P x*ð Þ , *t* of the Uhlenbeck-Ornstein process satisfies the

*∂x* ½ �þ ð Þ *x* � *x*0 *P D ∂*2

The fundamental solution of this linear parabolic partial differential equation,

2*Dτ*

*μ* ¼ *x*0 þ ð Þ *y* � *x*0 *e*

*σ*2 ¼ *Dτ* 1 � *e*

density function dependent on two parameters *μ* and *σ*, formally:

*p x*ð Þ¼ , *μ*, *σ* 1

ð Þ¼ *θ*1, *θ*2

*ψ θ*ð Þ¼ 1, *θ*2

where *x*0 represents the average length of the displacement, *D* is the diffusion coefficient, and *τ* is a characteristic time. Without loss of generality, in the rest of

Considering Eqs. (12) and (13), we think the function (Eq. (11)) as a probability

ffiffiffiffiffiffiffiffiffi 2*πσ*2 p *e*

Considering these parameters as the coordinates ð Þ *μ*, *σ* of the manifold *M* and taking into account Eqs. (2) and (4), it can be seen that for *α* ¼ 0 and *α* ¼ �1, the

Inspired by the simplicity of relations (Eqs. (5) and (6)), we use an alternative

1 2*σ*2 , � *μ σ*2

In these coordinates the probability density function (Eq. (14)) belongs to the

*p x*ð Þ¼ , *θ*1, *θ*2 exp �*θ*1*x*2 � *θ*2*x* � *ψ θ*ð Þ 1, *θ*2

*θ*2 2 4*θ*1

� ln

ffiffiffiffi *θ*1 *π* " # r

� �

�ð Þ *x*�*μ* 2 *=*2*σ*2

�2*t=τ* � �

and the initial condition consisting of a unit point mass at location *y*, is:

exp � 1

*P*

*x* � *x*0 � ð Þ *y* � *x*0 *e*�*t=τ* � �2 1 � *e*�2*t=τ*

( ) " #

*∂x*2 *:* (10)

�*t=τ* (12)

*:* (14)

*:* (15)

*:* (17)

� � (16)

(11)

(13)

Substituting Eqs. (22) and (23) in Eq. (20) for the coordinate *θ*1, we have

$$\frac{1}{2D\tau^2} \left\{ 2D\tau[(a-1)-(a-3)\coth(t/\tau)] - (a-1)\mathbf{x}\_0^2 \right\} = \frac{d^2\mathbf{t}/d\mathbf{s}^2}{\left(dt/d\mathbf{s}\right)^2},\tag{24}$$

and for the coordinate *θ*2

$$\begin{aligned} \frac{\text{csch}(t/\tau)}{2D\tau^2} \{2D\tau[-1-(a-2)\cosh\left(t/\tau\right)+(a-1)\sinh\left(t/\tau\right)]\}+\\ \frac{\text{csch}(t/\tau)}{2D\tau^2} \left[\left(1+e^{-t/\tau}\right)(a-1)\chi\_0^2\right] = \frac{d^2t/ds^2}{(dt/ds)^2}. \end{aligned} \tag{25}$$

The results (Eqs. (24) and (25)) are compatible if we choose *α* as

$$a = 1 + 2\frac{D\tau}{x\_0^2}.\tag{26}$$

In this case the characteristic time *τ* represents the tumbling time, and *ν* represents

*t as a function of s for two values of τ. τ* ¼ 10 *in the solid line and τ* ¼ 1 *in the dashed line.*

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

Thinking the geodesic curve as an optimal trajectory, from Eq. (21) we will analyze the optimal evolution of the system in terms of the behavior of the macroscopic potential (Eq. (17)) and the entropy function. To begin with this discussion, we first notice that the common definition of a nonequilibrium Gibbs entropy

suggests to define a trajectory-dependent entropy for the particle (or "system") *h x*ð Þ¼� , *θ* ln ½ � *p x*ð Þ , *θ* [3]. The trajectory-dependent entropy is related to the function *l x*ð Þ , *θ* used in the construction of the metric of the statistical manifold *M* by means of the relation *h x*ð Þ¼� , *θ l x*ð Þ , *θ* . In our analysis we are interested in the

Moreover, given that the probability density function (Eq. (16)) belongs to the exponential family, we can calculate the entropy by means of the relation [13].

> *∂ψ ∂θ*1 � *θ*2 *∂ψ ∂θ*2

> > ffiffiffiffi *θ*1 *π* " # r

*H*ð Þ¼ *θ ψ* � *θ*1

*H*ð Þ¼ *θ*

1 2 � ln

In order to study the temporal evolution of potential *ψ* and entropy *H*, we use the relations Eqs. (12) and (13) in Eq. (21). In particular, we focus on investigating the asymptotic behavior, that is, the behavior for *t* ! ∞. In this limit and using Eq. (21), we have that *θ*1 ¼ 1*=*ð Þ 2*Dτ* and *θ*2 ¼ �*x*0*=*ð Þ *Dτ* . Therefore, the asymptotic

*p x*ð Þ , *θ* ln ½ � *p x*ð Þ , *θ dx* ¼ h i *h x*ð Þ , *θ* (30)

*:* (31)

*:* (32)

the average speed of bacteria [12].

**Figure 1.**

**4. Optimal trajectory and entropy**

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

*H*ð Þ¼� *θ*

In any case, using Eqs. (30) or (31), we obtain

behavior of the potential and entropy is written as

**79**

entropy averaged over trajectories *H*ð Þ*θ* .

∞ð

�∞

Taking as reference the geometric study in the coordinates ð Þ *μ*, *σ* [7], we consider two possible relationships between the characteristic time *τ* and the other two parameters, *τ* ¼ *x*2 0*=D* or *τ* ¼ *x*2 0*=*ð Þ 2*D* . The first case leads to a flat geometry, that is, if *τ* ¼ *x*2 0*=D*, we have that *α* ¼ 3 and for Eq. (19) *R*ð Þ3 1212ð Þ¼ *θ*1, *θ*2 0. This case is equivalent to the connection *α* ¼ �1 in the coordinates ð Þ *μ*, *σ* , where the manifold is also flat and the affine time *s* and Newtonian time *t* are related by the differential equation

$$\frac{1}{\tau} = \frac{d^2 t / ds^2}{\left(dt / ds\right)^2} \tag{27}$$

whose solution has the form

$$t = -\tau \ln\left(s + a\tau\right) + b.\tag{28}$$

On the other hand if *τ* ¼ *x*2 0*=*ð Þ 2*D* , we have that *α* ¼ 2 and the affine and Newtonian time are now related by

$$\frac{\coth(t/\tau)}{\tau} = \frac{d^2t/ds^2}{\left(dt/ds\right)^2}.\tag{29}$$

Due to its nature, the equation (Eq. (29)) has been solved numerically. For each *τ* we find solutions that make sense in the context of our problem. In **Figure 1** we show two examples of these solutions.

In the coordinates ð Þ *μ*, *σ* , Eqs. (27) and (29) correspond to the choices *α* ¼ �1 and *α* ¼ 0, respectively [7]. On the other hand in the context of geometric construction from the potential (Eq. (17)), the main interpretation of the equation (Eq. (26)) is that the temporal evolution described by Eq. (21) coincides with a geodesic curve of space ð Þ *θ*1, *θ*2 . In this sense we find two relationships *τ* ¼ *x*2 0*=D* and *τ* ¼ *x*2 0*=*ð Þ 2*D* between the parameters *x*0, *D*, and *τ* for which the evolution is optimized.

It is interesting to note that if we assume that *v* ¼ *x*0*=τ*, the relationship *τ* ¼ *x*2 0*=D* leads us to *D* ¼ *v*2*τ*. This is a well-known and a widely used relationship, for example, in the understanding of bacterial mobility through a diffusive model.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

**Figure 1.** *t as a function of s for two values of τ. τ* ¼ 10 *in the solid line and τ* ¼ 1 *in the dashed line.*

In this case the characteristic time *τ* represents the tumbling time, and *ν* represents the average speed of bacteria [12].

#### **4. Optimal trajectory and entropy**

Thinking the geodesic curve as an optimal trajectory, from Eq. (21) we will analyze the optimal evolution of the system in terms of the behavior of the macroscopic potential (Eq. (17)) and the entropy function. To begin with this discussion, we first notice that the common definition of a nonequilibrium Gibbs entropy

$$H(\theta) = -\int\_{-\infty}^{\infty} p(\mathbf{x}, \theta) \ln \left[ p(\mathbf{x}, \theta) \right] d\mathbf{x} = \langle h(\mathbf{x}, \theta) \rangle \tag{30}$$

suggests to define a trajectory-dependent entropy for the particle (or "system") *h x*ð Þ¼� , *θ* ln ½ � *p x*ð Þ , *θ* [3]. The trajectory-dependent entropy is related to the function *l x*ð Þ , *θ* used in the construction of the metric of the statistical manifold *M* by means of the relation *h x*ð Þ¼� , *θ l x*ð Þ , *θ* . In our analysis we are interested in the entropy averaged over trajectories *H*ð Þ*θ* .

Moreover, given that the probability density function (Eq. (16)) belongs to the exponential family, we can calculate the entropy by means of the relation [13].

$$H(\theta) = \psi - \theta\_1 \frac{\partial \psi}{\partial \theta\_1} - \theta\_2 \frac{\partial \psi}{\partial \theta\_2}. \tag{31}$$

In any case, using Eqs. (30) or (31), we obtain

$$H(\theta) = \frac{1}{2} - \ln\left[\sqrt{\frac{\theta\_1}{\pi}}\right].\tag{32}$$

In order to study the temporal evolution of potential *ψ* and entropy *H*, we use the relations Eqs. (12) and (13) in Eq. (21). In particular, we focus on investigating the asymptotic behavior, that is, the behavior for *t* ! ∞. In this limit and using Eq. (21), we have that *θ*1 ¼ 1*=*ð Þ 2*Dτ* and *θ*2 ¼ �*x*0*=*ð Þ *Dτ* . Therefore, the asymptotic behavior of the potential and entropy is written as

Substituting Eqs. (22) and (23) in Eq. (20) for the coordinate *θ*1, we have

2*Dτ*2 f2*Dτ*½ � �1 � ð Þ *α* � 2 cosh ð Þþ *t=τ* ð Þ *α* � 1 sinh ð Þ *t=τ* gþ

*α* ¼ 1 þ 2

Taking as reference the geometric study in the coordinates ð Þ *μ*, *σ* [7], we consider two possible relationships between the characteristic time *τ* and the other two

equivalent to the connection *α* ¼ �1 in the coordinates ð Þ *μ*, *σ* , where the manifold is also flat and the affine time *s* and Newtonian time *t* are related by the differential

*t=ds*2

1 *τ* ¼ *d*2

cothð Þ *t=τ*

*τ* ¼ *d*2

Due to its nature, the equation (Eq. (29)) has been solved numerically. For each *τ* we find solutions that make sense in the context of our problem. In **Figure 1** we

In the coordinates ð Þ *μ*, *σ* , Eqs. (27) and (29) correspond to the choices *α* ¼ �1 and *α* ¼ 0, respectively [7]. On the other hand in the context of geometric construction from the potential (Eq. (17)), the main interpretation of the equation (Eq. (26)) is that the temporal evolution described by Eq. (21) coincides with a geodesic curve of space ð Þ *θ*1, *θ*2 . In this sense we find two relationships *τ* ¼ *x*2

0*=*ð Þ 2*D* between the parameters *x*0, *D*, and *τ* for which the evolution is

It is interesting to note that if we assume that *v* ¼ *x*0*=τ*, the relationship *τ* ¼

0*=D* leads us to *D* ¼ *v*2*τ*. This is a well-known and a widely used relationship, for example, in the understanding of bacterial mobility through a diffusive model.

¼ *d*2

*t=ds*2 ð Þ *dt=ds* 2 *:*

> *Dτ x*2 0

� � ¼ *d*2

0

*t=ds*2

*:* (26)

1212ð Þ¼ *θ*1, *θ*2 0. This case is

ð Þ *dt=ds* 2 (27)

ð Þ *dt=ds* 2 *:* (29)

0*=D*

*t* ¼ �*τ* ln ð Þþ *s* þ *aτ b:* (28)

0*=*ð Þ 2*D* , we have that *α* ¼ 2 and the affine and

*t=ds*2

0*=*ð Þ 2*D* . The first case leads to a flat geometry, that is,

ð Þ *dt=ds* 2 , (24)

(25)

2*Dτ*2 2*Dτ α*½ð Þ� � 1 ð Þ *α* � 3 cothð Þ *t=τ* � � ð Þ *α* � 1 *x*2

ð Þ *α* � 1 *x*2 0

The results (Eqs. (24) and (25)) are compatible if we choose *α* as

1

and for the coordinate *θ*2

2*Dτ*2 1 þ *e*

whose solution has the form

On the other hand if *τ* ¼ *x*2

Newtonian time are now related by

show two examples of these solutions.

�*t=τ* � �

*Advances on Tensor Analysis and Their Applications*

0*=D* or *τ* ¼ *x*2

0*=D*, we have that *α* ¼ 3 and for Eq. (19) *R*ð Þ3

h i

cschð Þ *t=τ*

cschð Þ *t=τ*

parameters, *τ* ¼ *x*2

if *τ* ¼ *x*2

equation

and *τ* ¼ *x*2

optimized.

*x*2

**78**

*Advances on Tensor Analysis and Their Applications*

$$\psi = \frac{\varkappa\_0^2}{2D\tau} + \frac{1}{2}\ln\left[2\pi D\tau\right] \tag{33}$$

**5. Discussion and perspectives**

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

steady state of an optimal evolution in a flat manifold.

(UNLP). AM, LL, and CR are professors at the UNLP.

*R*ð Þ 0

**81**

nonequilibrium systems.

**Acknowledgements**

In this chapter we have studied the PDF (Eq. (14)) that is the fundamental solution of the Fokker-Planck equation associated with the UO process. Our main interest was relating the geometric aspects of the process with the steady-state behavior. In our analysis we used the theoretical framework of the statistical manifold *M* with *α*-connections for two different coordinates ð Þ *μ*, *σ* and ð Þ *θ*1, *θ*2 [4, 7]. In the first case, there exist two interesting values of *α*, namely, *α* ¼ �1 and *α* ¼ 0, for which the process evolves on a geodesic of space ð Þ *μ*, *σ* with different values of *τ*. However, in the search for a simpler geometric construction, we find that for the coordinates ð Þ *θ*1, *θ*2 , we can define a macroscopic potential *ψ θ*ð Þ 1, *θ*2 , and the values of *α* that lead to the system evolving on a geodesic curve are *α* ¼ 3 and *α* ¼ 2. In our study we show that the connection *α* ¼ �1 for the coordinates ð Þ *μ*, *σ* corresponds to the connection *α* ¼ 3 for the coordinates ð Þ *θ*1, *θ*2 in the sense that both connections lead to a flat curvature and the same relationship between the parameters of the system. An important consequence of this behavior is that when the system evolves over geodesics, the macroscopic potential *ψ* and entropy *H* have the same functional dependence in the steady state. If we think of the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy describes the

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

Additionally and poorly developed in this chapter is the use of geometric aspects in the study of instabilities in nonequilibrium system. In equilibrium thermodynamics this information is contained in the scalar curvature of the manifold of equilibrium states [2]. For nonequilibrium problems, the instabilities are associated with the singularities or discontinuities of the curvature tensor [5, 15]. In the case of diffusive problems, the instabilities can be found by studying the singularities of

1212ð Þ *μ*, *σ* . In terms of diffusivity, we see that as it decreases, the system has different macroscopic behavior. An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [12]. From the perspective of the macroscopic potential *ψ*, the instabilities can be associated to singularities or discontinuities of *ψ*. In this sense, from a wider point of view, we consider that the potential *ψ* represents an alternative way to study the phase transitions in

This work was supported by the Universidad Nacional de La Plata, Argentina

and

$$H = \frac{1}{2} + \frac{1}{2} \ln\left[2\pi D\tau\right].\tag{34}$$

If we think that in the coordinates ð Þ *θ*1, *θ*2 , with *α* ¼ 3, the system evolves on a geodesic of a flat manifold (*R*ð Þ3 1212ð Þ¼ *θ*1, *θ*2 0), we have *τ* ¼ *x*2 0*=D*, and both the potential *ψ* and the entropy *H* tend to the same asymptotic value. In other words the entropy describes the asymptotic behavior of an optimal evolution:

$$
\Delta \varphi = H = \frac{1}{2} + \ln \sqrt{2 \pi D \tau}. \tag{35}
$$

In this last result, it can be seen that the asymptotic behavior depends fundamentally on the transport coefficient *D*. In this sense, the relevant variable in the description of the process is the diffusion coefficient *D*. It is interesting to note that the relations (Eq. (21)) are not invariant when we change *t* by �*t*. In this context, the evolution of the system is irreversible.

While in the coordinates ð Þ *θ*1, *θ*2 are two *α*-connections which leads to the system on a geodesic curve (*α* ¼ 3 and *α* ¼ 2), only *α* ¼ 3 makes that the macroscopic potential and entropy have the same functional dependence. It is interesting to note that for this choice of *α*, the manifold in the coordinates ð Þ *θ*1, *θ*2 is flat.

In the study of processes, the speed at which they occur is important. In this sense we will study the rate of change of potential:

$$\begin{split} \dot{\boldsymbol{\nu}} &= \frac{\mathbf{x}\_0^2}{4D\tau^2} \text{sech}\left(\frac{t}{2\tau}\right) + \frac{1}{2\tau} \left[\coth\left(\frac{t}{\tau}\right) - \mathbf{1}\right] = \\ &= \frac{1}{2\tau} \text{sech}\left(\frac{t}{2\tau}\right) + \frac{1}{2\tau} \left[\coth\left(\frac{t}{\tau}\right) - \mathbf{1}\right] \ge \mathbf{0}, \end{split} \tag{36}$$

and the entropy

$$\dot{H} = \frac{1}{2\pi} \left[ \coth\left(\frac{t}{\pi}\right) - 1 \right] \ge 0. \tag{37}$$

The equal sign in Eqs. (36) and (37) corresponds to the steady state [14]. In this regard, we associate the asymptotic behavior with the steady state, and we can indicate that the entropy describes the steady state of an optimal evolution.

From the perspective of the probability density function (PDF), when the diffusion coefficient takes small values, the distribution (Eq. (14)) diffuses from a uniform state to a sharp concentrated state, that is, from a stable state to an unstable state [15]. Although in this chapter we have not studied the behavior of the curvature tensor in the ð Þ *μ*, *σ* coordinates, these instabilities observed from the perspective of the PDF can be expressed in the singularity of the curvature tensor *R*ð Þ 0 1212ð Þ *μ*, *σ* . An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [5]. We also observe from Eq. (35) that for *D* ! 0, the entropy and the macroscopic potential have a singular behavior. The behavior that we observe both the tensor *R*ð Þ 0 1212ð Þ *μ*, *σ* and the potential *ψ* allows us to conjecture that the instabilities of the system can also be observed in the singularities or discontinuities of the macroscopic potential.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

#### **5. Discussion and perspectives**

*ψ* ¼ *x*2 0 2*Dτ* þ 1 2

*H* ¼ 1 2 þ 1 2

entropy describes the asymptotic behavior of an optimal evolution:

*ψ* ¼ *H* ¼ 1

If we think that in the coordinates ð Þ *θ*1, *θ*2 , with *α* ¼ 3, the system evolves on a

potential *ψ* and the entropy *H* tend to the same asymptotic value. In other words the

In this last result, it can be seen that the asymptotic behavior depends fundamentally on the transport coefficient *D*. In this sense, the relevant variable in the description of the process is the diffusion coefficient *D*. It is interesting to note that the relations (Eq. (21)) are not invariant when we change *t* by �*t*. In this context,

While in the coordinates ð Þ *θ*1, *θ*2 are two *α*-connections which leads to the system on a geodesic curve (*α* ¼ 3 and *α* ¼ 2), only *α* ¼ 3 makes that the macroscopic potential and entropy have the same functional dependence. It is interesting to note that for this choice of *α*, the manifold in the coordinates ð Þ *θ*1, *θ*2 is flat. In the study of processes, the speed at which they occur is important. In this

> 2*τ* � � þ 1 2*τ* coth *t*

2*τ* coth *t*

regard, we associate the asymptotic behavior with the steady state, and we can indicate that the entropy describes the steady state of an optimal evolution.

*τ* � � � 1 h i

The equal sign in Eqs. (36) and (37) corresponds to the steady state [14]. In this

From the perspective of the probability density function (PDF), when the diffusion coefficient takes small values, the distribution (Eq. (14)) diffuses from a uniform state to a sharp concentrated state, that is, from a stable state to an unstable state [15]. Although in this chapter we have not studied the behavior of the curvature tensor in the ð Þ *μ*, *σ* coordinates, these instabilities observed from the perspective of the PDF can be expressed in the singularity of the curvature tensor *R*ð Þ 0

An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [5]. We also observe from Eq. (35) that for *D* ! 0, the entropy and the macroscopic potential have a singular behavior. The behavior that we observe both

of the system can also be observed in the singularities or discontinuities of the

1212ð Þ *μ*, *σ* and the potential *ψ* allows us to conjecture that the instabilities

sech *t* 2*τ* � � þ 1 2*τ* coth *t*

*H*\_ ¼ 1

*τ* � � � 1 h i

*τ* � � � 1 h i ¼

≥0*:* (37)

(36)

1212ð Þ *μ*, *σ* .

≥0,

1212ð Þ¼ *θ*1, *θ*2 0), we have *τ* ¼ *x*2

2 þ ln ffiffiffiffiffiffiffiffiffiffiffi

and

geodesic of a flat manifold (*R*ð Þ3

*Advances on Tensor Analysis and Their Applications*

the evolution of the system is irreversible.

sense we will study the rate of change of potential:

*ψ*\_ ¼ *x*2 0 4*Dτ*2 sech *t*

> ¼ 1 2*τ*

and the entropy

the tensor *R*ð Þ 0

**80**

macroscopic potential.

ln 2½ � *πDτ* (33)

ln 2½ � *πDτ :* (34)

2*πDτ* p *:* (35)

0*=D*, and both the

In this chapter we have studied the PDF (Eq. (14)) that is the fundamental solution of the Fokker-Planck equation associated with the UO process. Our main interest was relating the geometric aspects of the process with the steady-state behavior. In our analysis we used the theoretical framework of the statistical manifold *M* with *α*-connections for two different coordinates ð Þ *μ*, *σ* and ð Þ *θ*1, *θ*2 [4, 7]. In the first case, there exist two interesting values of *α*, namely, *α* ¼ �1 and *α* ¼ 0, for which the process evolves on a geodesic of space ð Þ *μ*, *σ* with different values of *τ*. However, in the search for a simpler geometric construction, we find that for the coordinates ð Þ *θ*1, *θ*2 , we can define a macroscopic potential *ψ θ*ð Þ 1, *θ*2 , and the values of *α* that lead to the system evolving on a geodesic curve are *α* ¼ 3 and *α* ¼ 2. In our study we show that the connection *α* ¼ �1 for the coordinates ð Þ *μ*, *σ* corresponds to the connection *α* ¼ 3 for the coordinates ð Þ *θ*1, *θ*2 in the sense that both connections lead to a flat curvature and the same relationship between the parameters of the system. An important consequence of this behavior is that when the system evolves over geodesics, the macroscopic potential *ψ* and entropy *H* have the same functional dependence in the steady state. If we think of the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy describes the steady state of an optimal evolution in a flat manifold.

Additionally and poorly developed in this chapter is the use of geometric aspects in the study of instabilities in nonequilibrium system. In equilibrium thermodynamics this information is contained in the scalar curvature of the manifold of equilibrium states [2]. For nonequilibrium problems, the instabilities are associated with the singularities or discontinuities of the curvature tensor [5, 15]. In the case of diffusive problems, the instabilities can be found by studying the singularities of *R*ð Þ 0 1212ð Þ *μ*, *σ* . In terms of diffusivity, we see that as it decreases, the system has different macroscopic behavior. An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [12]. From the perspective of the macroscopic potential *ψ*, the instabilities can be associated to singularities or discontinuities of *ψ*. In this sense, from a wider point of view, we consider that the potential *ψ* represents an alternative way to study the phase transitions in nonequilibrium systems.

#### **Acknowledgements**

This work was supported by the Universidad Nacional de La Plata, Argentina (UNLP). AM, LL, and CR are professors at the UNLP.

*Advances on Tensor Analysis and Their Applications*

**References**

5135-5145

[1] Ruppeiner G. Thermodynamics: A Riemannian geometric model. Physical

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

[11] Zhang J. On monotone embedding in information geometry. Entropy. 2015;

Pagonabarraga I, Tailleur J. Arrested phase separation in reproducing bacteria creates a generic route to pattern formation. Proceedings of the National Academy of Sciences. 2010;

[13] Jaynes ET. Information theory and statistical mechanics. Physics Review.

[14] Kondepudi D, Prigogine I. Modern Thermodynamics from Heat Engines to Dissipative Structures. 2nd ed. United Kingdom: John Wiley & Sons; 2015

[15] Obata T, Hara H, Endo K. Differential geometry of non-equilibrium processes. Physical Review A. 1992;**45**:6997-7001

[12] Cates M, Marenduzzo D,

**17**:4485-4499

**107**:11715-11720

1957;**106**:620-630

[2] Ruppeiner G. Riemannian geometric approach to critical points: General theory. Physical Review E. 1998;**57**:

[3] Seifert U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Physical Review

[4] Amari S. Differential-geometrical methods in statistics. In: Lecture Notes in Statistics. Vol. 28. Heindelberg:

[5] Melgarejo A, Langoni L, Ruscitti C. Instability in bacterial populations and the curvature tensor. Physica A: Statistical Mechanics and its Applications. 2016;**458**:189-193

[6] Uhlenbeck G, Ornstein L. On the theory of the Brownian motion. Physics

[7] Ruscitti C, Langoni L, Melgarejo A.

Uhlenbeck–Ornstein process. Physica

[8] Brody D, Hook D. Information geometry in vapour-liquid equilibrium. Journal of Physics A: Mathematical and

[9] Amari S. Differential geometry of curved exponential families-curvatures

Review. 1930;**36**:823-841

Scripta. 2019;**94**:115010

Description of instabilities in

Theoretical. 2009;**42**:023001

and information loss. Annals of Statistics. 1982;**10**:357-385

[10] Oshima H, Obata T, Hara H. Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics. Journal of Physics A: Mathematical and General. 1999;**32**:

6373-6383

**83**

Review A. 1979;**20**:1608-1613

Letters. 2005;**95**:040602

Springer; 1985

#### **Author details**

Claudia B. Ruscitti1†, Laura B. Langoni2† and Augusto A. Melgarejo2 \*†

1 Facultad de Ciencias Exactas, Departamento de Matemática, UNLP, La Plata, Argentina

2 Facultad de Ingeniería, Departamento de Ciencias Básicas, UNLP, La Plata, Argentina

\*Address all correspondence to<EMAIL_ADDRESS>
† These authors are contributed equally.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

#### **References**

[1] Ruppeiner G. Thermodynamics: A Riemannian geometric model. Physical Review A. 1979;**20**:1608-1613

[2] Ruppeiner G. Riemannian geometric approach to critical points: General theory. Physical Review E. 1998;**57**: 5135-5145

[3] Seifert U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Physical Review Letters. 2005;**95**:040602

[4] Amari S. Differential-geometrical methods in statistics. In: Lecture Notes in Statistics. Vol. 28. Heindelberg: Springer; 1985

[5] Melgarejo A, Langoni L, Ruscitti C. Instability in bacterial populations and the curvature tensor. Physica A: Statistical Mechanics and its Applications. 2016;**458**:189-193

[6] Uhlenbeck G, Ornstein L. On the theory of the Brownian motion. Physics Review. 1930;**36**:823-841

[7] Ruscitti C, Langoni L, Melgarejo A. Description of instabilities in Uhlenbeck–Ornstein process. Physica Scripta. 2019;**94**:115010

[8] Brody D, Hook D. Information geometry in vapour-liquid equilibrium. Journal of Physics A: Mathematical and Theoretical. 2009;**42**:023001

[9] Amari S. Differential geometry of curved exponential families-curvatures and information loss. Annals of Statistics. 1982;**10**:357-385

[10] Oshima H, Obata T, Hara H. Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics. Journal of Physics A: Mathematical and General. 1999;**32**: 6373-6383

[11] Zhang J. On monotone embedding in information geometry. Entropy. 2015; **17**:4485-4499

[12] Cates M, Marenduzzo D, Pagonabarraga I, Tailleur J. Arrested phase separation in reproducing bacteria creates a generic route to pattern formation. Proceedings of the National Academy of Sciences. 2010; **107**:11715-11720

[13] Jaynes ET. Information theory and statistical mechanics. Physics Review. 1957;**106**:620-630

[14] Kondepudi D, Prigogine I. Modern Thermodynamics from Heat Engines to Dissipative Structures. 2nd ed. United Kingdom: John Wiley & Sons; 2015

[15] Obata T, Hara H, Endo K. Differential geometry of non-equilibrium processes. Physical Review A. 1992;**45**:6997-7001

**Author details**

Argentina

Argentina

**82**

Claudia B. Ruscitti1†, Laura B. Langoni2† and Augusto A. Melgarejo2

\*Address all correspondence to<EMAIL_ADDRESS>
† These authors are contributed equally.

*Advances on Tensor Analysis and Their Applications*

provided the original work is properly cited.

1 Facultad de Ciencias Exactas, Departamento de Matemática, UNLP, La Plata,

2 Facultad de Ingeniería, Departamento de Ciencias Básicas, UNLP, La Plata,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*†

Section 4

Advanced Topics of

Tensor Analysis

**85**

Section 4


**6.1 Chemical composition of core BAF37**

Unit I has relatively high Ca, Ni, Fe, and Zr concentrations; low contents of Si, K, Ti, Mn, and Rb; and low ratios of Mn/Fe, V/V + Ni, and Mg/Ca (**Figure 4**). Contrarily, a tendency through the increased Mn/Fe and Mg/Ca ratios suggested for the sediments belonging to units II to IV. Overall downcore characterization also reflects increasing values of Si, K, Ti, Fe, Ni, Zn, Ca, and Zr in units III and IV. These elements correlated oppositely with declining Ca values, indicating the carbonate dilution effect. Especially, the lowermost unit (Unit V) is characterized by the obvious fluctuations of the Ca, Fe, Si, K, Ti, Rb, and Fe. In addition to element concentrations, cross-correlations of V/Cr–U/Th, Ni/Co–U/Th, V/V + Ni–U/Th,

**Table 2.**

*Factor loads of 16 selected elements (bold numbers indicate high positive and high negative factor loadings).*

**Figure 4.**

**35**

*interface phase and long core sediments, jointly).*

*Concentrations of selected element and element oxides (in ppm) in BAF37 core sediments (in ppm),TOC% contents, and HI values in BAF37 sediments (blue-colored area represents values measured for sediment–water*

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea)*

*DOI: http://dx.doi.org/10.5772/intechopen.85589*

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea) DOI: http://dx.doi.org/10.5772/intechopen.85589*

#### **Figure 4.**

*Concentrations of selected element and element oxides (in ppm) in BAF37 core sediments (in ppm),TOC% contents, and HI values in BAF37 sediments (blue-colored area represents values measured for sediment–water interface phase and long core sediments, jointly).*

**6. Chemo-stratigraphy**

**Table 2.**

**34**

**6.1 Chemical composition of core BAF37**

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

Chemical characterization of the sediments (BAF-3B; BAF37; BS) investigated applying ICP-Ms analysis and revealed abundances of selected elements (Al, K, Ti, Zr, Rb, Fe, Mn, Ni, V, Cu, Pb, Zn, Mg, Ca, Na, P, Ba, Sr). A selected statistical method is also applied using geochemical data, namely, "factor analysis" (FA). Accordingly, eigenvalues of I and II factors were determined from 16 variables (**Table 2**). Congruent factor load values indicate either the same geological sources or element enrichment processes [26]. However, several element contents (Cu, V, Pb, Zn) were below the detection limits (*bdl*) in the sediment–water interface sediments (core: BAF-3B). These values were not applied for FA approaches.

Unit I has relatively high Ca, Ni, Fe, and Zr concentrations; low contents of Si, K, Ti, Mn, and Rb; and low ratios of Mn/Fe, V/V + Ni, and Mg/Ca (**Figure 4**). Contrarily, a tendency through the increased Mn/Fe and Mg/Ca ratios suggested for the sediments belonging to units II to IV. Overall downcore characterization also reflects increasing values of Si, K, Ti, Fe, Ni, Zn, Ca, and Zr in units III and IV. These elements correlated oppositely with declining Ca values, indicating the carbonate dilution effect. Especially, the lowermost unit (Unit V) is characterized by the obvious fluctuations of the Ca, Fe, Si, K, Ti, Rb, and Fe. In addition to element concentrations, cross-correlations of V/Cr–U/Th, Ni/Co–U/Th, V/V + Ni–U/Th,

*factor 1 factor 2 factor 1 factor 2 factor 1 factor 2*

**Cores BAF 3B BAF37 BS**

*Al* **0.9** 0.1 **0.7** 0.6 **0.7** 0.6 *Fe* **0.9** 0.0 **0.9** 0.3 **1.0** 0.2 *Mg* **1.0** 0.0 **0.9** 0.5 **0.8** 0.2 *Ca* 0.3 **0.9** 0.2 **0.8 0.7** 0.6 *Na* 0.7 0.7 0.4 0.6 **0.9** 0.4 *K* **0.9** 0.2 **0.9** 0.3 0.7 0.7 *Ti* **0.9** 0.3 **0.8** 0.6 **0.8** 0.5 *P* **0.7** 0.2 0.0 **0.9 0.9** 0.1 *Mn* **0.8** 0.5 0.7 0.2 **0.9** 0.2 *Ni* **0.9** 0.1 **0.8** 0.4 **0.9** 0.2 *Ba* **0.9** 0.3 **0.8** 0.3 0.5 **0.7** *Sr* **0.8** 0.3 0.7 0.4 0.5 0.6 *V bdl bdl* **0.8** 0.4 **0.9** 0.3 *Cu bdl bdl* 0.5 0.5 **0.9** 0.1 *Pb bdl bdl* 0.4 0.3 **0.9** 0.0 *Zn bdl bdl* 0.6 0.6 **0.9** 0.3 *Expl.Var* 8.4 1.8 7.5 4.3 10.7 2.9 *Prp.Totl* 0.7 0.2 0.5 0.3 0.7 0.2

*Factor loads of 16 selected elements (bold numbers indicate high positive and high negative factor loadings).*

**Figure 5.** *Redox conditions of the lake bottom waters and the adjacent swamp environment.*

and V/Cr–Ni/Co were applied for detailed investigations of the past lake water column properties (**Figure 5**).

#### **6.2 Chemical composition of the borehole (swamp) section (BS)**

Fluctuated values between average to higher values of Si, TiO2, and Zr elements are observed in sediments retrieved from the swamp section. These elements exhibit similar patterns along this section. Contrarily, CaO, MnO, FeO, and Rb concentrations reflect average to relatively low concentrations. Uppermost two units (Us I and Us II) contain relatively high CaO, K2O, and MnO contents but low to average concentrations of Si, Ti, Fe2O3, and Zr (**Figure 6**). The highest concentrations of Si, Zr, and TiO2 are observed in unit Us III, which has overall relatively low concentrations of CaO, MnO, FeO, and K2O. The lowermost unit (Us IV) also reflects a characteristic geochemical signature, with the uppermost part of this unit containing relatively high K2O, CaO, MnO, FeO, Zn, and Rb and low Si, TiO2, and Zr concentrations. The general downcore increasing tendency of Si, TiO2, and Zr concentrations continues in this unit. The rapid variability of the redox-sensitive elements (Zn, V) and V/V + Ni was observed for 2.6–4.7 m interval. Lower values of these elements and element ratios were observed for the 7.9–9.7 and 11.3–12 m intervals and at 6.2 m depth (**Figure 6**). In accordance to V/Cr, U/Th, Ni/Co, and V/V + Ni cross-correlations, BS sediments are placed at the same specific area, which is also characteristic for the BAF37 sediments (**Figure 5**).

#### **7. Discussion**

A quantitative chemostratigraphic approach, together with sedimentological observations, has been used for a better understanding of the Late Holocene paleoecological history of the Lake Bafa and related aquatic environments. Furthermore, a sum of the selected paleo-ecosystem parameters (e.g., element concentrations and element ratios) have been applied to identify the physical dynamics of the environment and the chemical characteristics of the water column, in terms of oxidation level and salinity. Furthermore, main environmental controls on Lake Bafa aquatic ecosystem were constructed using statistical approaches (FA) on geochemical data.

*Concentrations of selected elements and element oxides along the drill core BS, including TOC and HI data, and*

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea)*

*DOI: http://dx.doi.org/10.5772/intechopen.85589*

*specific element ratios as a measure for organic matter accumulation rates.*

**Figure 6.**

**37**

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea) DOI: http://dx.doi.org/10.5772/intechopen.85589*

#### **Figure 6.**

and V/Cr–Ni/Co were applied for detailed investigations of the past lake water

are observed in sediments retrieved from the swamp section. These elements exhibit similar patterns along this section. Contrarily, CaO, MnO, FeO, and Rb concentrations reflect average to relatively low concentrations. Uppermost two units (Us I and Us II) contain relatively high CaO, K2O, and MnO contents but low to average concentrations of Si, Ti, Fe2O3, and Zr (**Figure 6**). The highest concentrations of Si, Zr, and TiO2 are observed in unit Us III, which has overall relatively low concentrations of CaO, MnO, FeO, and K2O. The lowermost unit (Us IV) also reflects a characteristic geochemical signature, with the uppermost part of this unit containing relatively high K2O, CaO, MnO, FeO, Zn, and Rb and low Si, TiO2, and Zr concentrations. The general downcore increasing tendency of Si, TiO2, and Zr concentrations continues in this unit. The rapid variability of the redox-sensitive elements (Zn, V) and V/V + Ni was observed for 2.6–4.7 m interval. Lower values of these elements and element ratios were observed for the 7.9–9.7 and 11.3–12 m intervals and at 6.2 m depth (**Figure 6**). In accordance to V/Cr, U/Th, Ni/Co, and V/V + Ni cross-correlations, BS sediments are placed at the same specific area,

Fluctuated values between average to higher values of Si, TiO2, and Zr elements

**6.2 Chemical composition of the borehole (swamp) section (BS)**

*Redox conditions of the lake bottom waters and the adjacent swamp environment.*

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

which is also characteristic for the BAF37 sediments (**Figure 5**).

A quantitative chemostratigraphic approach, together with sedimentological observations, has been used for a better understanding of the Late Holocene paleoecological history of the Lake Bafa and related aquatic environments. Furthermore, a sum of the selected paleo-ecosystem parameters (e.g., element concentrations and

column properties (**Figure 5**).

**Figure 5.**

**7. Discussion**

**36**

*Concentrations of selected elements and element oxides along the drill core BS, including TOC and HI data, and specific element ratios as a measure for organic matter accumulation rates.*

element ratios) have been applied to identify the physical dynamics of the environment and the chemical characteristics of the water column, in terms of oxidation level and salinity. Furthermore, main environmental controls on Lake Bafa aquatic ecosystem were constructed using statistical approaches (FA) on geochemical data.

These controls include the external processes of terrigenous supply, leading to the enrichment of the siliciclastic elements, such as Si, Ti, and K and internal processes of biologic activity, which results in enrichment of Ca, Mg, Sr (in endogenic carbonate), TOC, and P. Herewith, high negative Factor I loads indicate similar geological sources for Al, K, Ti, Fe, Mg, Ni, Ba, and V enrichments for lacustrine sediments (cores BAF37 and BAF3B) (**Table 2**). Taking into consideration the Factors I and II loadings together, two main subgroups are suggested for these sediments. The first group is mainly a clastic-sourced element group (Al, Ti, Fe, Mg, K). The second group is related to the endogenic processes of carbonate deposition (Ca, Sr) and organic productivity (P) and water and sediment column redox processes (Ni, Co, V). The enrichment of these transition metals (Mn, Fe, Ni) is probably controlled by both detrital input and redox processes in the water and sediment columns. The factor load signature indicates that the swamp section reflects similar sources and modes of enrichment for most of these elements (**Table 2**). This suggests that the same geological processes prevailed also in the adjacent swamp area. However, there are also differences; Ba, Sr, and Ti enrichment pathways are different in the swamp sediments. Moreover, the nutrient elements (mainly P) and endogenic carbonate group elements (Ca and Sr) have similar factor (high Factor II and lower Factor I) loadings, suggesting a strong association with the organic productivity.

TOC concentrations). Phosphorus is known as the main nutrient marker, which would limit the biological productivity in any aquatic ecosystem [27, 32]. Ba enrichments are usually considered as a productivity proxy, being incorporated in the diatom frustules as micro-barite crystals [11, 27, 28]. However, these two elements would easily mobilize in anoxic conditions [33, 34]. Therefore, their signals may partly diminish from the sediments [33, 34]. Particularly, bulk organic carbon content (TOC) indicates changes in accumulation and/or preservation intensity under changing paleoecological conditions [10, 35–37]. Furthermore, hydrogen index (HI) values were applied to determine sedimentary organic matter sources

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea)*

*DOI: http://dx.doi.org/10.5772/intechopen.85589*

TOC content of lacustrine sediments is observed in a wide range between 0.7 and 4.4% (BAF37). TOC concentrations of swamp section sediments are quite low (≤1%) (**Figure 6**). Mean TOC content is 2.15% in the lake section and, however, 0.6% in swamp section. This obvious difference probably arises from overall distinct organic matter production processes, accumulation ranges, and preservation conditions. Main nutrient input parameter, P, is relatively low in the older lacustrine sediments (core BAF 37; >1.2 cal ka years). During the recent period, P availability was higher than the mean values. Therefore, P was probably not a limiting factor for organic matter productivity in Lake Bafa water column. Overall Ba trend which is similar to that of P, without obvious fluctuations, supports this

Detailed investigations of lacustrine sediments supply additional information about time-dependent organic matter accumulation rate signs. Therefore, organic matter transportation and production ranges were probably intensive during the accumulation stages of the older sediments (Units III, IV, and V). During this stage, the organic matter type was probably controlled by a mixed contribution of the aquatic and terrestrial sources, reflected by the relatively higher HI values. Only exceptions are observed in two (2.74 and 3.98 cm) thin layers. These layers have lower HI values, mainly suggesting terrestrial organic matter sources. Similarly, organic matter accumulation of the recent lacustrine sediments (Units I–II; last 0.8-ka-year record) and the entire swamp section (BS) are also sourced by the

Chemical properties of the water column are evaluated focusing on respective changes in salinity and redox conditions. Mg/Ca provides important evidence for the past salinity history of the water column. Basically, enhanced salinity (Mg/Ca) ratio favors the formation of aragonite, whereas low salinity (Mg/Ca) ratios promote calcite precipitation. This variation has important effects on the biota and the

The Mg/Ca trend in Lake Bafa's sedimentary successions indicates an upward decrease, which suggests a decrease in salinity in time (**Figure 7**). Time-dependent decrease of the lake water column chemistry from past to recent terms is also supported by the diatom analysis results of the same sedimentary section (core

type of carbonate mineral accumulated in the sediments [29, 39]. Minerals containing Ti are conservative in geochemical reactions (e.g., redox conditions) [40]. Contrarily, changing redox conditions strongly affect iron (Fe) and manganese (Mn) cycles either in sediments or water column (trend toward lower pE). In particularly, Mn easily mobilizes in changing redox conditions than Fe [11, 41]. Accordantly, low Mn/Fe ratios reflect reducing conditions. Following the similar aspects V/Cr, U/Th, Ni/Co, U/Th, and V/V + Ni, cross-correlations were used to

[37, 38].

conclusion (**Figure 4**).

**7.3 Water chemistry**

terrestrial vegetation supply (**Figure 7**).

interpret past redox conditions [11, 12, 42, 43].

BAF37), published by Bulkan et al. [8, 9].

**39**

#### **7.1 Detrital input and changes in hydrological conditions**

The interpretation of the detrital sources and transport intensity of the detrital material allows us to determine the physical dynamics and energy conditions of the Lake Bafa Basin. Basically, elemental enrichment of Si, K, Ti, Zr, and Rb indicates the terrigenous material supply. Contrarily Sr, Ca, Mg, and Ba reflect biogenic sources [11, 27–29]. K, Si, and Ti element enrichments and enhanced average grain size distribution of the lake sediments (BAF37), "accumulated during the period of 2.5–2.2 ka year BP," indicate deposition under high energy conditions (**Figure 4**). This likely corresponds to a period when intensive freshwater input occurred during the earlier stages of separation of Lake Bafa from the Aegean Sea. However, this input was interrupted abruptly and followed by a short-term low energy conditions, indicated by the increasing clay size fraction and Ca (carbonate) contents of the sediments [8, 13, 31]. Average detrital input was low during the period of 1.95–0.8 cal. ka year BP, except for a brief period around 1.8 cal. ka year BP when relatively high energy conditions prevailed, which was likely caused by an abrupt hydrological change. After 0.8 cal. ka BP, the lake became completely isolated from the sea [8, 9] but continued to be influenced by the water and sediment inputs in its western part from the Büyük Menderes River. This river flooding events strongly influenced the Lake Bafa hydrology and caused abrupt water level fluctuations as well as rapid increases in the sedimentation rates during the Late Holocene [8, 9, 13, 30, 31]. The high variability of the river discharge was mainly controlled by the climate-driven rainfall pattern. During the enhanced discharge events, transported sediments could easily reach the lake since the streambed slopes have lower gradients. Therefore, we would suggest the climate-driven processes mainly controlled the variability of the hydrological conditions in the isolated lake during the last 800 or so years. Starting from the 1990s, a rubber dam was constructed. Since then, the lake level is currently artificially controlled.

#### **7.2 Organic matter productivity**

Relative variations of the biological production rates were determined, applying organic matter accumulation indicators (i.e., P concentrations, Ba enrichments,

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea) DOI: http://dx.doi.org/10.5772/intechopen.85589*

TOC concentrations). Phosphorus is known as the main nutrient marker, which would limit the biological productivity in any aquatic ecosystem [27, 32]. Ba enrichments are usually considered as a productivity proxy, being incorporated in the diatom frustules as micro-barite crystals [11, 27, 28]. However, these two elements would easily mobilize in anoxic conditions [33, 34]. Therefore, their signals may partly diminish from the sediments [33, 34]. Particularly, bulk organic carbon content (TOC) indicates changes in accumulation and/or preservation intensity under changing paleoecological conditions [10, 35–37]. Furthermore, hydrogen index (HI) values were applied to determine sedimentary organic matter sources [37, 38].

TOC content of lacustrine sediments is observed in a wide range between 0.7 and 4.4% (BAF37). TOC concentrations of swamp section sediments are quite low (≤1%) (**Figure 6**). Mean TOC content is 2.15% in the lake section and, however, 0.6% in swamp section. This obvious difference probably arises from overall distinct organic matter production processes, accumulation ranges, and preservation conditions. Main nutrient input parameter, P, is relatively low in the older lacustrine sediments (core BAF 37; >1.2 cal ka years). During the recent period, P availability was higher than the mean values. Therefore, P was probably not a limiting factor for organic matter productivity in Lake Bafa water column. Overall Ba trend which is similar to that of P, without obvious fluctuations, supports this conclusion (**Figure 4**).

Detailed investigations of lacustrine sediments supply additional information about time-dependent organic matter accumulation rate signs. Therefore, organic matter transportation and production ranges were probably intensive during the accumulation stages of the older sediments (Units III, IV, and V). During this stage, the organic matter type was probably controlled by a mixed contribution of the aquatic and terrestrial sources, reflected by the relatively higher HI values. Only exceptions are observed in two (2.74 and 3.98 cm) thin layers. These layers have lower HI values, mainly suggesting terrestrial organic matter sources. Similarly, organic matter accumulation of the recent lacustrine sediments (Units I–II; last 0.8-ka-year record) and the entire swamp section (BS) are also sourced by the terrestrial vegetation supply (**Figure 7**).

#### **7.3 Water chemistry**

These controls include the external processes of terrigenous supply, leading to

The interpretation of the detrital sources and transport intensity of the detrital material allows us to determine the physical dynamics and energy conditions of the Lake Bafa Basin. Basically, elemental enrichment of Si, K, Ti, Zr, and Rb indicates the terrigenous material supply. Contrarily Sr, Ca, Mg, and Ba reflect biogenic sources [11, 27–29]. K, Si, and Ti element enrichments and enhanced average grain size distribution of the lake sediments (BAF37), "accumulated during the period of 2.5–2.2 ka year BP," indicate deposition under high energy conditions (**Figure 4**). This likely corresponds to a period when intensive freshwater input occurred during the earlier stages of separation of Lake Bafa from the Aegean Sea. However, this input was interrupted abruptly and followed by a short-term low energy conditions, indicated by the increasing clay size fraction and Ca (carbonate) contents of the sediments [8, 13, 31]. Average detrital input was low during the period of

1.95–0.8 cal. ka year BP, except for a brief period around 1.8 cal. ka year BP when relatively high energy conditions prevailed, which was likely caused by an abrupt hydrological change. After 0.8 cal. ka BP, the lake became completely isolated from the sea [8, 9] but continued to be influenced by the water and sediment inputs in its western part from the Büyük Menderes River. This river flooding events strongly influenced the Lake Bafa hydrology and caused abrupt water level fluctuations as well as rapid increases in the sedimentation rates during the Late Holocene [8, 9, 13, 30, 31]. The high variability of the river discharge was mainly controlled by the climate-driven rainfall pattern. During the enhanced discharge events, transported sediments could easily reach the lake since the streambed slopes have lower gradients. Therefore, we would suggest the climate-driven processes mainly controlled the variability of the hydrological conditions in the isolated lake during the last 800 or so years. Starting from the 1990s, a rubber dam was constructed. Since then, the

Relative variations of the biological production rates were determined, applying organic matter accumulation indicators (i.e., P concentrations, Ba enrichments,

the enrichment of the siliciclastic elements, such as Si, Ti, and K and internal processes of biologic activity, which results in enrichment of Ca, Mg, Sr (in endogenic carbonate), TOC, and P. Herewith, high negative Factor I loads indicate similar geological sources for Al, K, Ti, Fe, Mg, Ni, Ba, and V enrichments for lacustrine sediments (cores BAF37 and BAF3B) (**Table 2**). Taking into consideration the Factors I and II loadings together, two main subgroups are suggested for these sediments. The first group is mainly a clastic-sourced element group (Al, Ti, Fe, Mg, K). The second group is related to the endogenic processes of carbonate deposition (Ca, Sr) and organic productivity (P) and water and sediment column redox processes (Ni, Co, V). The enrichment of these transition metals (Mn, Fe, Ni) is probably controlled by both detrital input and redox processes in the water and sediment columns. The factor load signature indicates that the swamp section reflects similar sources and modes of enrichment for most of these elements (**Table 2**). This suggests that the same geological processes prevailed also in the adjacent swamp area. However, there are also differences; Ba, Sr, and Ti enrichment pathways are different in the swamp sediments. Moreover, the nutrient elements (mainly P) and endogenic carbonate group elements (Ca and Sr) have similar factor (high Factor II and lower Factor I) loadings, suggesting a strong

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

association with the organic productivity.

lake level is currently artificially controlled.

**7.2 Organic matter productivity**

**38**

**7.1 Detrital input and changes in hydrological conditions**

Chemical properties of the water column are evaluated focusing on respective changes in salinity and redox conditions. Mg/Ca provides important evidence for the past salinity history of the water column. Basically, enhanced salinity (Mg/Ca) ratio favors the formation of aragonite, whereas low salinity (Mg/Ca) ratios promote calcite precipitation. This variation has important effects on the biota and the type of carbonate mineral accumulated in the sediments [29, 39]. Minerals containing Ti are conservative in geochemical reactions (e.g., redox conditions) [40]. Contrarily, changing redox conditions strongly affect iron (Fe) and manganese (Mn) cycles either in sediments or water column (trend toward lower pE). In particularly, Mn easily mobilizes in changing redox conditions than Fe [11, 41]. Accordantly, low Mn/Fe ratios reflect reducing conditions. Following the similar aspects V/Cr, U/Th, Ni/Co, U/Th, and V/V + Ni, cross-correlations were used to interpret past redox conditions [11, 12, 42, 43].

The Mg/Ca trend in Lake Bafa's sedimentary successions indicates an upward decrease, which suggests a decrease in salinity in time (**Figure 7**). Time-dependent decrease of the lake water column chemistry from past to recent terms is also supported by the diatom analysis results of the same sedimentary section (core BAF37), published by Bulkan et al. [8, 9].

variety of terrestrial aquatic ecosystems, ranging from fluvial, lacustrine, and lagoonal to coastal marine environments [6, 47–55]. Additional factors affecting the environmental conditions in such systems include abrupt geological events (e.g., storms, earthquakes, and tsunamis) and global fluctuations of sea level changes related to orbital climatic oscillations. We applied both sedimentological and geochemical proxies to unravel the time-dependent stages of landscape evolution in the

*"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea)*

Several geological and geomorphological studies have been carried out in and around the Lake Bafa These studies have contributed to our knowledge of the coastal geomorphological history, sea level changes, tectonic processes driven erosion rates, climate and environmental induced changes (vegetation and faunal

One study by Bruckner et al. [48] suggests that horst and graben tectonics in the Büyük and Küçük Menderes Grabens caused serious environmental and coastline changes. Mullenhoff et al. [6] concluded that the Lake Bafa was formed as a consequence of erosional processes in the adjacent mainland of Turkey, which have controlled the deltaic progradation and the filling of the Latmian Gulf. Particularly, in Miletus and the Büyük Menderes Graben, remarkable transformations have been revealed, with the metamorphosis of the marine gulf into residual lakes (e.g., lakes Azap and Bafa) [48]. The coastline of the Büyük Menderes Delta, located close to the southern graben area, progradaded seaward some 5 km [5] during the last 1.5 ka cal. year BP. Further progradation closed the entrance of the Latmian Gulf that resulted in the isolation of the Lake Bafa from the Aegean Sea at 0.8 cal. ka year BP. Lake Marmara in the Gediz Graben in the north, with a similar setting to that of the Lake Bafa, shows resembling environmental evolution [5, 7, 47], with the detrital input records reflecting synchronous changes within its catchment area. Similar to the Lake Bafa, Marmara witnessed a marked environmental change with a shift toward more oxic and freshwater conditions at 0.95 cal ka year BP and the

The strongest human impact has been detected at the time of the Greek period in the seventh- to first-century BC and especially during the Roman period in the firstcentury BC until the fourth-century AD, when sedimentation was about five times higher than that in the periods before and after [22]. These changes were accompanied by changes in the vegetation type. The palynological analyses have shown high amounts of Quercus type before the period of the strong human impact, which is also reflected by the element composition of the relatively recent sediments, water column samples, plant, and shells that have been analyzed for the pollution assess-

A sum of local environmental conditions in the Lake Bafa and its catchment changed during the last 4.5 cal. ka years, but still several questions in this context remain to be answered. Basic questions are as follows: (I) Are the sediment accumulation rates different during the lacustrine, lagoon, or shallow water environmental phases and transitional stages? [53, 54], (II) how does the radiocarbon reservoir effect varies with changing environmental stages and time? [53, 54], (III) what is the temporal evolution of meandering delta progradation? and (IV) would it be possible to estimate the behavior and frequency of the river flood intensity from the lake's sedimentary record? These questions can be addressed by acquiring and analyzing additional long cores from different parts of the lake and from the delta. Furthermore, the same cores can be used to study rapid geological and climatic events such as tsunamis, volcanic explosions, and storms. Additional methods, including diatom analysis [8, 9], isotope analysis [8, 9], and lipid or amino acid biomarker analyses, would contribute further to our understanding of the

increased effects of fluvial activity during the last 300 years. [47].

coastal area of the Büyük Menderes Graben.

*DOI: http://dx.doi.org/10.5772/intechopen.85589*

changes) and human impact [6, 8, 9, 48–57].

ment studies [49–52].

**41**

ecological evolution of this unique basin.

#### **Figure 7.**

*Element ratio indicator for water chemistry conditions. (a) represents specific element ratios for core BAF37 (blue-colored area represents values measured for sediment–water interface phase and long core sediments, jointly). (b) indicates drill core BS sediments.*

Previous studies suggest that Cr, U, and V elements are rare in denitrifying conditions. Contrarily, enhanced Ni, Co, Cu, Zn, Cd, and Mo contributions are documented for organic-rich sediments, accumulated under sulfate-reducing bottom water conditions [11, 44–46]. Cross-correlations of V/Cr–U/Th, Ni/Co**–**U/Th, V/V + Ni**–**U/Th, and V/Cr**–**Ni/Co indicate that both the lake and the swamp sections deposited mainly under oxic conditions (**Figures 5** and **7**). Cr, U, and V element ratios in the Lake Bafa core and the swamp section indicate deposition under mainly oxic conditions. Despite most of the redox proxies showing the general oxic water column conditions, Ni/Co ratio points to somehow anoxic conditions. Furthermore, relatively low Mn/Fe ratio of the lake sediments (core BAF-37) indicates that the water column was probably partly oxygen depleted during 1.7–1.4 cal. ka year BP and during the isolated lake period, starting ca. 800 years ago.

#### **7.4 Implications for Holocene age local environmental conditions**

Both tectonics and sedimentological processes (N-S extension, graben, horst, erosion, delta progradation) and climate events (both orbital and abrupt changes) have played spectacular roles in modulating the ecological conditions in the Eastern Mediterranean Sea coastal areas. The ecological systems in this region include a

#### *"Geo-archives of a Coastal Lacustrine Eco-system": Lake Bafa (Mediterranean Sea) DOI: http://dx.doi.org/10.5772/intechopen.85589*

variety of terrestrial aquatic ecosystems, ranging from fluvial, lacustrine, and lagoonal to coastal marine environments [6, 47–55]. Additional factors affecting the environmental conditions in such systems include abrupt geological events (e.g., storms, earthquakes, and tsunamis) and global fluctuations of sea level changes related to orbital climatic oscillations. We applied both sedimentological and geochemical proxies to unravel the time-dependent stages of landscape evolution in the coastal area of the Büyük Menderes Graben.

Several geological and geomorphological studies have been carried out in and around the Lake Bafa These studies have contributed to our knowledge of the coastal geomorphological history, sea level changes, tectonic processes driven erosion rates, climate and environmental induced changes (vegetation and faunal changes) and human impact [6, 8, 9, 48–57].

One study by Bruckner et al. [48] suggests that horst and graben tectonics in the Büyük and Küçük Menderes Grabens caused serious environmental and coastline changes. Mullenhoff et al. [6] concluded that the Lake Bafa was formed as a consequence of erosional processes in the adjacent mainland of Turkey, which have controlled the deltaic progradation and the filling of the Latmian Gulf. Particularly, in Miletus and the Büyük Menderes Graben, remarkable transformations have been revealed, with the metamorphosis of the marine gulf into residual lakes (e.g., lakes Azap and Bafa) [48]. The coastline of the Büyük Menderes Delta, located close to the southern graben area, progradaded seaward some 5 km [5] during the last 1.5 ka cal. year BP. Further progradation closed the entrance of the Latmian Gulf that resulted in the isolation of the Lake Bafa from the Aegean Sea at 0.8 cal. ka year BP. Lake Marmara in the Gediz Graben in the north, with a similar setting to that of the Lake Bafa, shows resembling environmental evolution [5, 7, 47], with the detrital input records reflecting synchronous changes within its catchment area. Similar to the Lake Bafa, Marmara witnessed a marked environmental change with a shift toward more oxic and freshwater conditions at 0.95 cal ka year BP and the increased effects of fluvial activity during the last 300 years. [47].

The strongest human impact has been detected at the time of the Greek period in the seventh- to first-century BC and especially during the Roman period in the firstcentury BC until the fourth-century AD, when sedimentation was about five times higher than that in the periods before and after [22]. These changes were accompanied by changes in the vegetation type. The palynological analyses have shown high amounts of Quercus type before the period of the strong human impact, which is also reflected by the element composition of the relatively recent sediments, water column samples, plant, and shells that have been analyzed for the pollution assessment studies [49–52].

A sum of local environmental conditions in the Lake Bafa and its catchment changed during the last 4.5 cal. ka years, but still several questions in this context remain to be answered. Basic questions are as follows: (I) Are the sediment accumulation rates different during the lacustrine, lagoon, or shallow water environmental phases and transitional stages? [53, 54], (II) how does the radiocarbon reservoir effect varies with changing environmental stages and time? [53, 54], (III) what is the temporal evolution of meandering delta progradation? and (IV) would it be possible to estimate the behavior and frequency of the river flood intensity from the lake's sedimentary record? These questions can be addressed by acquiring and analyzing additional long cores from different parts of the lake and from the delta. Furthermore, the same cores can be used to study rapid geological and climatic events such as tsunamis, volcanic explosions, and storms. Additional methods, including diatom analysis [8, 9], isotope analysis [8, 9], and lipid or amino acid biomarker analyses, would contribute further to our understanding of the ecological evolution of this unique basin.

Previous studies suggest that Cr, U, and V elements are rare in denitrifying conditions. Contrarily, enhanced Ni, Co, Cu, Zn, Cd, and Mo contributions are documented for organic-rich sediments, accumulated under sulfate-reducing bottom water conditions [11, 44–46]. Cross-correlations of V/Cr–U/Th, Ni/Co**–**U/Th, V/V + Ni**–**U/Th, and V/Cr**–**Ni/Co indicate that both the lake and the swamp sections deposited mainly under oxic conditions (**Figures 5** and **7**). Cr, U, and V element ratios in the Lake Bafa core and the swamp section indicate deposition under mainly oxic conditions. Despite most of the redox proxies showing the general oxic water column conditions, Ni/Co ratio points to somehow anoxic conditions. Furthermore, relatively low Mn/Fe ratio of the lake sediments (core BAF-37) indicates that the water column was probably partly oxygen depleted during 1.7–1.4 cal. ka year BP

*Element ratio indicator for water chemistry conditions. (a) represents specific element ratios for core BAF37 (blue-colored area represents values measured for sediment–water interface phase and long core sediments,*

and during the isolated lake period, starting ca. 800 years ago.

**Figure 7.**

**40**

*jointly). (b) indicates drill core BS sediments.*

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

**7.4 Implications for Holocene age local environmental conditions**

Both tectonics and sedimentological processes (N-S extension, graben, horst, erosion, delta progradation) and climate events (both orbital and abrupt changes) have played spectacular roles in modulating the ecological conditions in the Eastern Mediterranean Sea coastal areas. The ecological systems in this region include a


Management of Acute and Chronic Pancreatitis

*Tommaso Stecca, Cristina Nistri, Bruno Pauletti, Adriana Di Giacomo, Flavio Colaut, Mariangela Ruperto, Ezio Caratozzolo, Luca Bonariol and Marco Massani*

## **Abstract**

Pancreatitis is a major public health issue worldwide. There is geographical variation in the burden of acute and chronic pancreatitis (CP). Globally, the agestandardized prevalence rate increased from 1990 to 2017. Acute pancreatitis (AP) is now one of the most common reasons for hospitalization with a gastrointestinal condition. The essential requirements for the management of AP are accurate diagnosis, appropriate triage, high-quality supportive care, monitoring for and treatment of complications, and prevention of relapse. Clinicians should be aware of the time course and the best management of AP, identifying which patient will have a severe course allowing earlier triage to an intensive care unit and earlier initiation of effective therapy. CP is a pathologic fibroinflammatory syndrome of the pancreas in individuals with genetic, environmental, and other risk factors who develop persistent pathologic responses to parenchymal injury or stress. Diagnosing the underlying pathologic process early in the disease course and managing the syndrome to change the natural course of disease and minimize adverse disease effects are the managing paradigm. In this review, we consider recent changes in the management of acute and CP, as well as common misunderstandings and areas of ongoing controversy.

**Keywords:** acute pancreatitis, chronic pancreatitis, management, clinical phases, pathologic process

## **1. Introduction**

Pancreatitis is a major public health issue worldwide. There is geographical variation in the burden of acute and chronic pancreatitis (CP). Globally, the agestandardized prevalence rate increased from 1990 to 2017. Acute pancreatitis (AP) is now one of the most common reasons for hospitalization with a gastrointestinal condition. The essential requirements for the management of AP are accurate diagnosis, appropriate triage, high-quality supportive care, monitoring for and treatment of complications, and prevention of relapse. Clinicians should be aware of the time course and the best management of AP, identifying which patient will have a severe course allowing earlier triage to an intensive care unit and earlier initiation of effective therapy. CP is a pathologic fibroinflammatory syndrome of the pancreas in individuals with genetic, environmental, and other risk factors who develop persistent pathologic responses to parenchymal injury or stress. Diagnosing the underlying pathologic process early in the disease course and managing the syndrome to change the natural course of disease and minimize adverse disease effects is the managing paradigm. In this review, we consider recent changes in the management of acute and CP, as well as common misunderstandings and areas of ongoing controversy.

Current concepts of the use of interventional methods in severe acute, necrotizing, and CP (indications and timing of interventions, strategies for intervention, endoscopic and percutaneous treatment) are discussed in the other chapters of this monograph on pancreatitis. We, therefore, consider it appropriate that they are illustrated in detail in the respective chapters.


**9. Conclusion**

'Domestication is that process by which a population of animals becomes adapted to man and to captive environment by genetic changes occurring over generation and environmentallyinduced developmental events recurring in each generation [90]'. It affects all functions of the organisms and, in particular, behaviour. There are behavioural differences between wild and reared fishes (see **Table 1**), but these differences are more quantitative than qualitative; no new behaviours appear with domestication.

**Behavioural traits Type of responses to**

Swimming behaviour Response to a predator attack

Foraging behaviour

Predator avoidance

Personality

**environmental constraints**

Capacity to be released in

Changing the foraging

Reaction in front of a

Reaction to a potential

Spontaneous escape

Social behaviour Aggressiveness Hatchery is more aggressive

Choice of place to spawn

Cannibalism No information?

Response to a novel environment

the wild

strategy

predator

predator

responses

Reproduction Abilities to reproduce after realising

**Effects of domestication References**

Wild exhibits lower dispersal [16, 17, 18]

Wild is more rapid [41, 42]

Hatchery takes more risks [36, 45]

It is higher for hatchery [55, 58]

It is higher for wild [56, 65] There is no difference [57]

Dominance is higher in wild [73]

Paternity rate higher for wild [79]

Better for wild [76] No difference [77]

Wild higher than for wild [82]

[10]

Effects of Domestication on Fish Behaviour http://dx.doi.org/10.5772/intechopen.78752

[19, 20]

[21, 22]

[29]

[47]

[56, 57]

[54]

[11, 12, 13, 14]

101

Wild is more reactive than

Wild exhibits higher swimming

Hatchery exploits all the water

Hatchery has a higher capture

No difference between wild

hatchery

abilities

column

Prey profitability Wild has better rate than hatchery [26]

and hatchery

than wild

Use of shelter Wild has a higher rate [47, 48]

Agonistic behaviour It is higher for hatchery [58, 63, 64]

Competitive behaviour Territoriality is higher in wild [70, 72]

Fitness in natural sites Higher for wild [71] Boldness Higher for hatchery [72, 80]

acquisition

Stress Wild is more stressful than hatchery [83, 84]

QTL mapping There is no difference in prey

survival

rate

Consequences on survival Wild shows a higher rate of

The selection of individuals for economic reasons leads to the selection of fishes on morphological or developmental traits (growth, size). These traits are directly linked to other biological traits (i.e. behaviour) and their selection may lead to select fishes, which present some behaviours affecting the life in groups of high density and so the development of each individual (increase in aggressive and agonistic interactions between individuals, higher levels of cannibalism). One solution to prevent that is to identify as soon as possible in the fish development the behavioural profiles of the individuals under different domestication levels [91]. These studies lead to better knowledge of the fish larvae, which are difficult to test given their high sensitivity to environmental conditions.

In this review, we saw that all behavioural traits may be impacted by domestication even after only one generation. For some traits, the results are clear and follow the same trend; the response to a predator is affected by domestication whatever the domesticated species and the reared environment. But in some cases, it is more difficult to find a common trend: foraging is affected but it depends on the type of food, and on the feeding conditions. It is the same for aggressiveness in the hatchery-reared individuals; it could decrease in that way we can put a high number of predators together if we give them a sufficient amount of food, but on the other hand, the high fish density in tank can produce a high level of aggressiveness between individuals leading to cannibalism event if the food is abundant. It is also true for other behavioural traits such as personality or cognitive capabilities; until now, there is a lack of studies on the influence of domestication on these behavioural traits and it is not possible to conclude. What we know is that the human selection on morphological or physiological traits of some individuals (even through a genetic program) has a direct influence on behavioural traits.

This has two implications: first, it is necessary to study behavioural traits in the case of domestication of new species in order to determine the best environmental conditions of rearing, and second, these behavioural trait modifications must be into account when release of domesticated animals into natural habitats is considered. For these two points, we have to keep in mind that the consequences of behavioural selection traits through domestication correspond to the selection of a particular behavioural trait belonging to the natural behavioural range of the species under rearing environmental conditions; this might lead to a new species, the other behavioural traits of the species range disappearing. It is known under a genetic-environment process by which the epigenetic landscape is modified by the environment constraints influencing directly the genetic program [92, 93].

**9. Conclusion**

100 Animal Domestication

traits.

new behaviours appear with domestication.

their high sensitivity to environmental conditions.

'Domestication is that process by which a population of animals becomes adapted to man and to captive environment by genetic changes occurring over generation and environmentallyinduced developmental events recurring in each generation [90]'. It affects all functions of the organisms and, in particular, behaviour. There are behavioural differences between wild and reared fishes (see **Table 1**), but these differences are more quantitative than qualitative; no

The selection of individuals for economic reasons leads to the selection of fishes on morphological or developmental traits (growth, size). These traits are directly linked to other biological traits (i.e. behaviour) and their selection may lead to select fishes, which present some behaviours affecting the life in groups of high density and so the development of each individual (increase in aggressive and agonistic interactions between individuals, higher levels of cannibalism). One solution to prevent that is to identify as soon as possible in the fish development the behavioural profiles of the individuals under different domestication levels [91]. These studies lead to better knowledge of the fish larvae, which are difficult to test given

In this review, we saw that all behavioural traits may be impacted by domestication even after only one generation. For some traits, the results are clear and follow the same trend; the response to a predator is affected by domestication whatever the domesticated species and the reared environment. But in some cases, it is more difficult to find a common trend: foraging is affected but it depends on the type of food, and on the feeding conditions. It is the same for aggressiveness in the hatchery-reared individuals; it could decrease in that way we can put a high number of predators together if we give them a sufficient amount of food, but on the other hand, the high fish density in tank can produce a high level of aggressiveness between individuals leading to cannibalism event if the food is abundant. It is also true for other behavioural traits such as personality or cognitive capabilities; until now, there is a lack of studies on the influence of domestication on these behavioural traits and it is not possible to conclude. What we know is that the human selection on morphological or physiological traits of some individuals (even through a genetic program) has a direct influence on behavioural

This has two implications: first, it is necessary to study behavioural traits in the case of domestication of new species in order to determine the best environmental conditions of rearing, and second, these behavioural trait modifications must be into account when release of domesticated animals into natural habitats is considered. For these two points, we have to keep in mind that the consequences of behavioural selection traits through domestication correspond to the selection of a particular behavioural trait belonging to the natural behavioural range of the species under rearing environmental conditions; this might lead to a new species, the other behavioural traits of the species range disappearing. It is known under a genetic-environment process by which the epigenetic landscape is modified by the environ-

ment constraints influencing directly the genetic program [92, 93].

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**Table 1.** Summary of the different results found for this review. The behavioural traits were divided in behavioural acts and their responses to domestication process in fish were briefly described. Wild referred to animals coming from wild strains and hatchery for fish larvae reared under farm or laboratory conditions.