The extension of thermodynamics that has reference to the rates of physical processes is the theory of non-equilibrium, or irreversible thermodynamics. Linear irreversible thermodynamics is based on the postulates of thermodynamics plus the additional postulate of time reversal symmetry of the laws of physics. This additional postulate states that the laws of physics remain unchanged if timet is everywhere replaced by its negative−t and if simultaneously the magnetic fieldB is replaced by its negative −B. Non-equilibrium thermodynamics is based on the Onsager Reciprocity Theorem, formulated by Lars Onsager. Statistical theorems like, the fluctuation-dissipation theorem, the Kubo relations, and the formalism of the linear response theory also exist. In this section we will briefly review these theories.
2.2.1 Notion of Local Equilibrium
A physical system can be so far out of equilibrium that quantities like temperature and pressure cannot be even defined. On the other hand, there are many cases where one can define these quan- tities. We consider an idealized case where we assume the system to be composed of homogeneous cells, small on the macroscopic scale but large on the microscopic one. We also assume that cells interact weakly with their neighbors so that each cell independently attains a local equilibrium with a microscopic relaxation time, τmicro, which is very small compared to the macroscopic relaxation time,τmacro, needed to achieve global equilibrium. The local equilibrium is said to exist when
1. Each subsystem is at equilibrium independently of the other subsystems.
2. Interactions between neighboring subsystems are weak.
When the system is in local equilibrium, all we need is the local conservation of energy and particles, and the local equation of state. The local equation of state equates the rate of entropy production,
˙
s, with the fluxJi and the corresponding affinity (driving force):
˙ s=X
i
FiJi. (2.6)
The second step involves the use of phenomenological relations between the affinities Fi and the corresponding fluxesJk. For this we need to invoke the famous Onsager’s reciprocal theorem. We will discuss that in the next section.
2.2.2 Onsager’s Reciprocal Relations
For certain systems the fluxes at a given instant depend only on the values of the affinities at that instant. Such systems are referred to as resistive systems. For a purely resistive system, by definition, each local flux depends only upon the instantaneous local affinities and the local intensive parameters. That is, dropping the indices denoting vector components,
Jk =Jk(F0, ...,Fj, ...;F0, ..., Fj, ...). (2.7) Though it is true that each flux Ji tends to depend most strongly on its own associated affinity Fi, its dependence on other affinities is the source of the most interesting phenomena in the field of irreversibility. Each flux Ji is known to vanish as the affinities vanish, so we can expandJk in powers of the affinities with no constant term:
Jk =X
j
LjkFj+ 1 2!
X
i
X
j
LijkFiFj+... (2.8)
where
Ljk= ∂Jk
∂Fj
0
(2.9)
and
Lijk=
∂2Jk
∂Fi∂Fj
0
. (2.10)
. The functions Ljk are called the kinetic coefficients. They are functions of the local extensive parameter. For the purposes of the Onsager theorem, it is convenient to adopt a notation that exhibits the functional dependence of the kinetic coefficients on an externally applied magnetic field
B, suppressing the dependence of other intensive parameters:
Ljk=Ljk(B). (2.11)
The Onsager theorem states that
Ljk(B) =Lkj(−B), (2.12)
that is, the value of the kinetic coefficientLjkmeasured in an external magnetic fieldBis identical to the value ofLkjmeasured in the reversed magnetic fieldB. The Onsager theorem states a symmetry between the linear effect of thejth affinity on thekth flux and the linear effect of thekth affinity on thejth flux when these effects are measured in opposite magnetic fields. If the affinities are so small that all quadratic and higher order terms in Eq. 2.8 can be neglected and
Jk =X
j
LjkFj (2.13)
such a process is called a linear purely resistive process. Onsager theorem is a powerful tool for the analysis of such a process.
In this thesis, a simple derivation of Onsager’s reciprocal theorems (derived for a more general case by E. T. Jaynes [7, 8]) by the application of maximum entropy methods to a simple model is explained in Chapter 7. The derivation is really simple, and makes one appreciate the beauty and elegance of the maximum entropy methods.
2.2.3 Fluctuation-dissipation Theorem, Green-Kubo Relations, and Lin- ear Response Theory
Consider a system in equilibrium with extensive macroscopic parameters A1, ...Am. Now, if we perturb this system from equilibrium by disturbing the parameter Ai by δAi = Ai− hAii, the perturbation on the Hamiltonian will be of the form
∆H =−X
j
fjAj, (2.14)
so that the equilibrium Hamiltonian will be transformed to H →H1=H0+ ∆H =H−X
j
fjAj. (2.15)
Thefj’s are often called the external forces, or simply the forces. Just as a matter of convention, we denoteAias the average of the quantityAiusing the modified Hamiltonian (i.e., the modified prob- ability distribution function), while we usehAiito imply average using the equilibrium probability distribution function. For small values of forces it can be shown that [3]
δAj(t) = Ai(t)− hAii,
= βX
j
fjhδAi(t)δAj(0)i. (2.16)
The summand in Eq. 2.16 is called the Kubo function and is represented as
Cij(t) =hδAi(t)δAj(0)i. (2.17)
The Kubo function is directly related to the dynamical susceptibility ξij(t), which is defined by writing the most general formula for the dynamical linear response to an external time-dependent perturbationP
jfj(t)Aj:
δAi(t) =X
j
Z t
−∞
dt0ξij(t−t0)fj(t0). (2.18)
In Fourier space, and supposing that ξij(t−t0) vanishes for t0 > t, the convolution in Eq. 2.18 is transformed into a product
δAi(ω) =X
j
ξij(ω)fj(ω). (2.19)
Using a few manipulations it can be shown that
ξij(t) =−βC˙ij(t). (2.20)
This is most general version of the Kubo formula. This relationship between the equilibrium fluctua- tions and the dynamic susceptibility can be used to give relationships between transport coefficients and the equilibrium correlation functions. These relationships are called Green-Kubo formulas.
Using the analyticity properties of the susceptibility function and the the notion of causality that ξ(t) = 0 ift <0, which reflects the obvious requirement that effect must follow cause, one obtains what is called the fluctuation-dissipation theorem. The classical version of this theorem is
ξ00(ω) =1
2βωC(ω), (2.21)
whereξ00(t) is defined as
ξ00(t) = i
2βC(t).˙ (2.22)
The right hand side (RHS) of this equation is clearly the fluctuation part. It is a priori not clear that the left hand side (LHS) is the dissipation, but it can be shown that it indeed is the dissipation.
The linear response theory and the fluctuation-dissipation theorem provide the microscopic basis of Onsager relations, while the Green-Kubo relations give a method to obtain the transport coefficients for non-equilibrium processes by observing the non-equilibrium fluctuations in the system.