Theories of Fluctuations and Dissipation

This research monograph presents a mathematical approach based on stochastic calculus which tackles the "cutting edge" in porous media science and engineering - prediction of dispersivity from covariance of hydraulic conductivity (velocity). The problem is of extreme importance for tracer analysis, for enhanced recovery by injection of miscible gases, etc. This book explains a generalised mathematical model and effective numerical methods that may highly impact the stochastic porous media hydrodynamics. The book starts with a general overview of the problem of scale dependence of the dispersion coefficient in porous media. Then a review of pertinent topics of stochastic calculus that would be useful in the modeling in the subsequent chapters is succinctly presented. The development of a generalised stochastic solute transport model for any given velocity covariance without resorting to Fickian assumptions from laboratory scale to field scale is discussed in detail. The mathematical approaches presented here may be useful for many other problems related to chemical dispersion in porous media.

Any description of a process once expressed in mathematical abstraction, becomes a "contracted" description or a contracted model. What is important to understand is that different levels of contracted descriptions could be useful for different purposes, and at the same time, the insights gained from one level of description should be obtained by another level of description and vice versa. However, this is a very difficult task in many of the molecular level processes. One of the main reasons for this difficulty is that the most of the molecular processes are thermodynamically irreversible. In addition, physics of the processes at different levels of descriptions are based on different conceptual frameworks, albeit being very meaningful at a given level. In our discussion here, we consider the thermodynamic level of description, the Boltzmann level of description, and physical ensemble description which is inherently stochastic, hence described in stochastic processes.

Thermodynamic Description
To facilitate the discussion here, we make use of the Brownian motion as an example with the aim of developing a general framework for discussion. The total differential of entropy of an idealized system of the Brownian particles can be written as, where U is the internal energy; V is the volume of the system; N is the number of particles (molecules); P is the pressure;  is the chemical potential; and, T is the absolute temperature. Equation (5.2.1) is a statement for a system of molecules and the system has the well-defined physical boundaries through which mass and heat transfer could occur. The momentum of the particles is included in the internal energy term, and by including the momentum ( M ) , the total energy is , where m is the mass of a particle. We can write equation (5.2.1) in the following form after including the momentum as a thermodynamic variable: .
In equation (5.2.2), v is the row vector of particle velocities and M is the column vector of particle moments. The total differential of entropy   dS can be expressed in terms of partial derivatives: Equation (5.2.10) can be derived from applying the Onsager principle for the two solids separately and taking into the account of the fact that the energy loss of one solid is the energy gained of the other solid. L in equation (5.2.10) is non-negative and needs to be determined experimentally. Further, equation (5.2.10) is only valid in the vicinity of the equilibrium.
The extensive variables, the momentum and the internal energy, are expressed as thermodynamic rule laws in equation (5.2.9) and (5.2.10); however, the momentum and the internal energy have quite distinct forms of functional characteristics. For example, if we reverse the velocities of the molecules, the magnetic field and the time associated with a physical ensemble, the momentum changes the direction but the internal energy remains the same. As the time progresses in the reverse direction, the ensemble will move along the past trajectory. when L is dependent on the external magnetic filed, B , when either the effects of the external magnetic field are ignored or the magnetic field is absent, the even or odd variables are coupled by a symmetric matrix where as the odd and even variables are coupled by a antisymmetric matrix. Equation (5.2.11) are called the Onsager-Casimir reciprocal relations.
To simplify the notation in the linear laws such as equation (5.2.5), we introduce the following variables: 1. In the linear laws, the thermodynamic forces are descriptions of the entropy of the system. At thermodynamic equilibrium, the entropy of a given system is maximum as the Second Law of thermodynamics says that entropy increases on the average of any spontaneous process. Let us consider the entropy of an isolated system in the vicinity of its thermodynamic equilibrium. The extensive variable a as defined before has finite values, and the entropy associated with the system, ( ) S a , can be expressed in terms of Taylor series: (5.2.14) at the maximum of ( ) S a , as these conditions are true for any a , the matrix ij S must be negative semi-definite. By incorporating equation (5.2.12) in equation (5.2.14), and using the conditions for the thermodynamic equilibrium stated previously, we obtain, But as an approximation, we can write, This can be expressed on the average Using equation (5.2.12), As the derivative of the entropy with respect to time is positive, L is a positive semidefinite matrix. What equation (5.2.17) and (5.2.18) convey is that the mean fluctuations of an extensive variable give rise to increase in the entropy, and therefore ( ) dS a dt alludes to dissipation of energy due to the fluctuations of an extensive variable. We define the Rayleigh-Onsager dissipation function as The dissipation function  is also called the entropy production and is dependent on our choice of the system.
Using equation ( where 0 a is the initial value of the selected process. The matrix H must be negative semi-definite as the entropy increases on the average during the relaxation process. Using equation (5.2.22), we can deduce the covariance function, As discussed before any dissipative system would have fluctuations in extensive variables. Let us define the fluctuations with reference to the expected value conditioned upon the initial value as, Computational Modelling of Multi-Scale Non-Fickian Dispersion in Porous Media -An Approach Based on Stochastic Calculus 168 to develop the simplest form of a stochastic differential equation. We will address this in section 5.4.

The Boltzmann Picture
As mentioned in the previous section, the Onsager regression hypothesis is based on the entropy and the coefficients which form the coupling matrix, L . The Boltzmann equation is on the other hand dependent entirely on the molecular dynamics of collisions and the resulting fluctuations. We do not intent to derive the Boltzmann equation here; instead, we describe the equation and the variables here. For the technical details of the derivation, there are many excellent texts on statistical mechanics and some of the original works are given in the references.
Boltzmann's work was on the dynamics of dilute gases and the average behaviour of gas molecules was the main focus on his work. The Boltzmann's equation describes the nonlinear dynamics of the molecular collisions, while Onsager theory is on linear dynamics without fluctuations. It can be shown that the linearized Boltzmann equation is a special case of Onsager theory.
In the derivation of the Boltzmann equation, we have a six-dimensional space in which the position of, r , and the velocity, v , of the centre of mass of a single molecule are defined.

Theories of Fluctuations and Dissipation 169
We call this six-dimensional space the -space or molecule phase space. We can divide the six-dimensional space into small cellular volumes and each volume elements is assigned an index 1, 2, 3, i   as a unique number for identification purposes. The number of molecules, ( ) i N t would be the macroscopic Boltzmann variable associated in the volume element i , and we choose the volume element i to be sufficiently large that ( ) i N t is a large number.
It is assumed that binary collisions between the molecules of only two volume elements located at , Then the Boltzmann equation gives, , F is an external force field acting in the -space, and r  and v  are the derivatives with respective to r and v , respectively. The third term on the right hand side of equation ( Unlike the Onsager's linear laws, which are true only near the thermodynamic equilibrium, the Boltzmann equation is true not only in the vicinity of the equilibrium but also away from the equilibrium. However, near the equilibrium these two pictures are similar and while the Boltzmann equation is valid strictly speaking only for diluted gases, the Onsager linear laws are valid for any ensemble. where ' e ' indicates the equilibrium state.

Boltzmann's theory includes a function called the H-function which behaves in an entropy
In the vicinity of equilibrium, we can write, is a small change in the -space density. By substituting equation (5.3.5) in the Boltzmann equation and ignoring the higher order terms of r D , we obtain, replacing the dissipation integral as a linear functional.
It can be shown that (Fox and Uhlenbeck, 1970 a and b) by adopting the Onsager hypothesis,   ( , , ), L is a linear operator (see Fox and Uhlenbeck, 1970 a and b); and f  is a random term which needs to be characterised.
The random term now can be defined by, In equation (5.3.7), the rate of change of the -space density increments are expressed in terms of thermodynamic forces   X .
By deriving the random term f  as in (5.3.8), we see that the random term is a zero-mean stochastic process in the -space,  -correlated in r and t but influenced by the velocity of the centre of mass through a linear operator derived from the dissipation term, S d , in the

Onsager Regression Hypothesis, Langevin Equation and Itō processes
The Onsager regression hypothesis, equation (5.2.26), states that the fluctuations of extensive variables around their expected values conditional on the initial values can be expressed in terms of a system of differential equations through a relaxation matrix which is defined in equation (5.2.20). Equation (5.2.26) is similar in form to equations (5.3.6) and (5.3.7) which are derived from Boltzmann's equation (5.3.1). Both of these theories support the hypothesis that the time derivatives of fluctuations on the average follow differential equations with additive random terms. The average fluctuations are driven by thermodynamically coupled driving forces because of energy dissipation according to Boltzmann. We have seen in the previous section that both of these descriptions are phenomenologically equivalent. However, none of those descriptions are amenable for operational models of fluctuation and dissipation.
Starting point of the development of such models is the Langevin equation which describes the motion of Brownian particles. Even though Langevin used the Newtonian laws to describe the particle motion, he developed a differential equation with an addictive random term, which is quite similar to the Onsager regression hypothesis. Langevin started by considering a particle of mass m at a distance r from an initial point, if p  is the momentum vector of the particle and V is the velocity, we can write from the Newton laws, We have slightly changed the notation to indicate that the variables are associated with a particle rather than with an ensemble.
In equation (5.4.2), F is the force vector on the particle (the particle is bombarded by the surrounding water molecules). We can express the F as ( ) In equations (5.4.5) and (5.4.1), the position of the particle, r and the momentum, p  are coupled, and f is a random additive noise. In a dissipative system, the random forcing term f can be assumed to have an expected value of zero: At the given time, the random force term, f at time 1 t is uncorrelated to that of 2 t , and it is a result if molecular impacts on the particle. Therefore, we can assume that f to be a  -correlated function: where 2  is the variance.
It is now clear that the Wiener process described in Chapter 2 is a good model for f , and therefore we can write equation (5.4.5) as, (5.4.6) where, ( ) w t is the standard Wiener process.
In the absence of an external force, Therefore, the solution of the stochastic differential equation (5.4.7), can be written as, (5.4.8) and equation (5.4.8) in an stochastic integral. The last integration can be interpreted in two ways: as an Itō integral or as a Stratonovich integral. Because of the martingale of property of Itō integrals, we choose to interpret the second integral on the right hand side of equation (5.4.8) as an Itō integral. The implications of this choice is important to understand: it makes stochastic processes such as equation (5.4.8) Markov processes with the transitional conditional probabilities obeying Fokker-Planck type equations. The stochastic differential equations of the type given by equation (5.4.7) describe the time evolution of stochastic variables. We generalize the stochastic differential of a vector valued stochastic process by, where n is an extensive variable, h is a vector function of n and t ,  is a diagonal matrix with ii  as the diagonal element, ( , ) g n t is a matrix function of n and t and w is the standard Wiener process vector. By taking  to be diagonal matrix, we assume that www.intechopen.com where V is the Gausssian velocity of the ensemble, V is the mean velocity and  is the "average" noise representing the fluctuations.
We have shown that the velocity can be expressed as consisting of a mean component and an additive fluctuating component, based on the Langevin description of Brownian particles. From an application point of view, the additive form of the velocity can be used to explain the local heterogeneity of the porous medium, i.e., we can always calculate the average velocity in a region and then the changes in the porous structure may be assumed to cause the fluctuations around the mean. This is the working assumption on which the stochastic solute transport model (SSTM) in Chapter 3 is based.

Thermodynamic Character of SSTM
As we have seen in section 5.3, equation (5.3.7) unites the Onsager and Boltzmann pictures close to equilibrium (Keizer, 1987). The SSTM given by equation (4.2.1) has a similar form to that of equation (5.3.7) and equation (5.3.2) where the fluctuating component is separated out as an additive component but the fluctuating part is now more complicated reflecting the influence of the porous media. According to equation (5.3.8), the "noisy" random functions have zero means and the two-time covariances are δ-correlated in time and space; and these Dirac's delta functions are related through a linear operator. In the development of SSTM, we assume only the δ-correlation in time because the spatial aspect is separated into a continuous function of space. This assumption can be justified as the porous medium influencing the fluctuations can be considered as a continuum.
www.intechopen.com This research monograph presents a mathematical approach based on stochastic calculus which tackles the "cutting edge" in porous media science and engineering -prediction of dispersivity from covariance of hydraulic conductivity (velocity). The problem is of extreme importance for tracer analysis, for enhanced recovery by injection of miscible gases, etc. This book explains a generalised mathematical model and effective numerical methods that may highly impact the stochastic porous media hydrodynamics. The book starts with a general overview of the problem of scale dependence of the dispersion coefficient in porous media. Then a review of pertinent topics of stochastic calculus that would be useful in the modeling in the subsequent chapters is succinctly presented. The development of a generalised stochastic solute transport model for any given velocity covariance without resorting to Fickian assumptions from laboratory scale to field scale is discussed in detail. The mathematical approaches presented here may be useful for many other problems related to chemical dispersion in porous media.