Non-Equilibrium Statistical Mechanics of Electrical Conduction in a Fermi Gas R. P. Rastogi and Mool C. Gupta Gorakhpur University, Gorakhpur, India Stimulated by the pioneering work ( I ) of Professor Prigogine and his school in Brussels, considerable interest has been generated in the field of non-equilibrium satistical mechanics. Formal solutions of the Liouville equation (2) in terms of a greenian and complete internal propagator has been shown by Kubo (3)to lead to the following expression (4) for electrical conductivity tensor o,
where p = IlkT; k is the Boltzmann constant and T is the temperature in Kelvin. (3,3,(~))0isthe microscopic current auto-correlation a t eauilibrium. J , denotes the averaee value of the microsn,pic current at time t whereas T id the time interval. Equation ( 1 ) is perfectly general and does not involve any model or appruximation. The two-time correlation functions are difficult to estimate hoth from theorv and experiment although some significant attempts have bken made to estimate these for molecular motions in liquids (5). The purpose of this note is to show that the classical formula for electrical conductivity of metals based on the free electron model can be ohtained from eqn. (1) in a straightforward manner. This shows that nonequilibrium statistical mechania has reached a stage when more meaningful results are expected to be ohtained in future. Solution of the Liouville Equation For the benefit of the readers, we will briefly outline the approach used for the solution of the Liouville equation since basic principles of non-equilibrium statistical mechanics are not widely known. The state of a system a t a given time t is defined by a distribution function F(p,q,t) where p and q denote the momenta and positional coordinates, respectively, of all the species constituting the system. If there are N species, the number of such canonical variables would be 2N. The Liouville equation is written as atF(t)= LF(t)
(2)
where the Liouvillian operator L is defined as follows
,=1
and
N
H' = X
N
E
V,.
j=*n-l
where V;. is the interaction between articles i and n a n d the summati& is w e r the total o f N par;icles. ~ ~ u a t i fur o n H' is obtained when uainvisearlditi\~itvisanumed. HF isassociated with the action of an externai field of potential VF such that 822
Journal of Chemical Education
where Zje is the charge on particle j. When L = Lo, we get the unperturbed Liouville equation. On the other hand when L = Lo + L' or when L = LO LF, we get the perturbed Liouville equation. Let us first consider the unperturbed Liouville equation,
+
The formal solution of the initial value problem can be written in the form (10) F(t) = UO(t)F(0) with the constraint that operator Uo(t) must satisfy the Liouville equation (11) d,U0(t) = LOUO(t) with the initial condition: UO(0) = I where I is the identity operator. Time-differentiation of both left-hand side and right-hand side shows that eqn. (10) is the solution of eqn. (9). Equation (10) defines the unperturbed propagator U"(t) which can also be expressed as Now, let us consider the unperturbed Liouville equation when L = Lo LF since this would hold good during electrical conduction in a Fermi gas. The solution of the Liouville equation can still be written as
+
d,W(t) = LW(t)
HO represents the free motion of non-interacting particles in the absence of an external field and is simply equivalent to the kinetic energy. H' represents the interaction between the particles so that
" ,x pjz/2mj
where the significance of the three components is obvious. We now consider a system consisting of several species of charged particles such that the total charge of the system is zero. For such a case
where the operator W(t) can he similarly defined by
H denotes the Hamiltonian and is given hy
H0 =
Since the Liouvillian operator L depends linearly on the Hamiltonian, we can break up the former corresponding t o different component of Liouvillian operator as follows
(14)
with the initial condition W(0) = I = Identity operator. However, in this case, we cannot assign aspecificmeaning to operator W(t) since although, W(t) = exp(tL) = exp(t(LO+LF)
(15)
we cannot give a precise meaning since the operators LO and LF cannot commute with each other. As a first steo it can be suggested that the following would be a solution bf the differential eqn. (14) W(t) = WO(t)+
S,'drWo(t - T)LW(T)
(16)
That this is so can he proved easily by the time-differentiation
of both the left-hand and right-hand side of eqn. (16). Ohviously, eqn. (16) is not a complete solution. In the zeroth approximation we can write The equation of motion for a free electron in an applied field E is given as
As a first approximation, the solution can he written as
m.~i=-e.E
(28)
so that Writing WO(t) = U(t), we can express the first-order solution to the Liouville equation as follows Fi1)(t)= Fo +
S,'~ ~ U ( T ) L ~ F O
(19)
since U(t)FO = FOand U(t - 7)FU= FObecause of the fact that equilibrium distribution is a stationary solution of the Liouville equation. Now, the electrical current J e ( t ) at time t is given by J'(t)
= JdNz
