## Abstract

The method of Riemannian geometry is fruitful in equilibrium thermodynamics. From the theory of fluctuations it has been possible to construct a metric for the space of thermodynamic equilibrium states. Inspired by these geometric elements, we will discuss the geometric-differential approach of nonequilibrium systems. In particular we will study the geometric aspects from the knowledge of the macroscopic potential associated with the Uhlenbeck-Ornstein (UO) nonequilibrium process. Assuming the geodesic curve as an optimal path and using the affine connection, known as α-connection, we will study the conditions under which a diffusive process can be considered optimal. We will also analyze the impact of this behavior on the entropy of the system, relating these results with studies of instabilities in diffusive processes.

### Keywords

- nonequilibrium processes
- Uhlenbeck-Ornstein process
- statistical manifold
- α-connections
- macroscopic potential

## 1. Introduction

The use of Riemannian geometry associated with the space of thermodynamic equilibrium states has been successful. In this framework the geometric elements are constructed from the knowledge of the thermodynamic potential. In this sense, one of the most interesting ideas is associated with the study of the phase transitions visualized by means of the singularities of the scalar curvature [1, 2]. As a geometric-differential approach of nonequilibrium systems, we consider in our study the geometric properties of a statistical manifold associated with trajectory-dependent entropy [3]. In statistics mechanics it is known that any statistical system has an associated metric-affine manifold having a special affine connection whether it is in equilibrium or not. The affine connection, called 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 two-dimensional metric-affine manifold in the coordinates

However, due to simplicity in geometric construction, we are interested in studying the behavior of the system in coordinates

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.

## 2. Elements of statistical manifold

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

becomes an

where

In the statistical manifold

We now restrict our attention to a special family of probability density function, called an exponential family, which is described by [8].

where

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].

These coefficients, for

where

In the case

From Eq. (3) the curvature tensor for an

Since

For systems in thermodynamic equilibrium, the parameters

## 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

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

which is the Gaussian density function with mean

and variance

where

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

Considering these parameters as the coordinates

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

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

where

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))

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

From Eq. (19) we observe that there are two

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)

with an arbitrary parameter

The tangent vector coordinate

and

Substituting Eqs. (22) and (23) in Eq. (20) for the coordinate

and for the coordinate

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

Taking as reference the geometric study in the coordinates

whose solution has the form

On the other hand if

Due to its nature, the equation (Eq. (29)) has been solved numerically. For each

In the coordinates

It is interesting to note that if we assume that

## 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

suggests to define a trajectory-dependent entropy for the particle (or “system”)

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].

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

In order to study the temporal evolution of potential

and

If we think that in the coordinates

In this last result, it can be seen that the asymptotic behavior depends fundamentally on the transport coefficient

While in the coordinates

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:

and the entropy

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

## 5. Discussion and perspectives

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

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

## Acknowledgments

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