N-particle distribution function

These two expressions are exact they allow us in principle to calculate the N-particle distribution function at time t (to the first order in the external field) if its initial value is known. This will be our starting point for analyzing electrolytes both at equilibrium and out of equilibrium. 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

Now return to the system of N particles, the distribution function for which, pN, changes frequently because of a large number of successive instantaneous collisions. Each collision causes the N-particle distribution function to change, because of the change of position and velocity of the two hard spheres which collide. The effect of all these collisions is additive and each instantaneously alters the distribution, pN. The Liouville equation for hard spheres is... 

The n-particle distribution function p(n ) in Eq. (10.41) is often replaced by the so-called potential of mean force function Wmean for the activated complex, defined as... 

Before we derive a relation between the rate constants in solution and gas phase, let us first verify that the expression for the rate constant in Eq. (10.40) simplifies to the well-known result for the gas-phase rate constant, when there is no solvent present. With no solvent, we have 14oi = 0. Also V t = 0, since this is the interaction energy between solvent and activated complex. The n-particle distribution function in Eq. (10.41) simplifies to... 

We note that the second term in (108) is the familiar Kirkwood expression for the stress tensor in terms of the n-particle distribution function. 

The approximate solution to the Schrodinger equation, defined by the effective Hamiltonian in Eq. (9-1), with either method described in the previous section, associates to every vector of molecule coordinates, R, together with the solvent-solvent interaction potential, an energy (R). From basic classical statistical mechanics an N-particle distribution function (PDF) n(R) is thus obtained ... 

Outside the ideal equilibrium conditions discussed, the number of parameters necessary to specify the coagulation process is large. STEPANOV [2.10] has attempted to develop a theory of coagulation in nonequilibrium systems starting with an equation of continuity for the N-particle distribution function f ( 2i >... 

We will usually call D ixi,. ..,Xn,t) the N-particle distribution function, in accordance with a notation to be developed as we proceed. 

If we are to derive the Boltzmann equation from the first hierarchy equation we must try to find the circumstances under which we can express the two-particle distribution function F2(xi,X2,t) appearing in Eq. (172) in terms of the single-particle function Fi(jCi, t). We will see in the next section that if the initial N-particle distribution function has certain properties, and if the density of the system is sufficiently low that noa can be regarded as a small expansion parameter, then we can express F2(xi, X2, t) in terms of Fi(jc, t) as a power series in and proceed to derive the Boltzmann equation and its generalization to higher order in the density. 

There are a number of different formulations of the time correlation function method, all of which lead to the same results for the linearized hydrodynamic equations. One way is to generalize the Chapman-Enskog normal solution method so as to apply it to the Liouville equations, and obtain the N-particle distribution function for a system near a local equilibrium state. " Expressions for the heat current and pressure tensor for a general fluid system can be obtained, which have the form of the macroscopic linear laws, with explicit expressions for the various transport coefficients. These expressions for the transport coefficients have the form of time integrals of equilibrium correlation functions of microscopic currents, viz., a transport coefficient t is given by... 

The Liouville equation is the basic equation of non-equilibrium statistical mechanics. It gives the time dependence of the N-particle distribution function f (X> )> derived... 

The expression for the pair (n = 2) distribution function in three (d = 3) dimension is well known [1,2]. However, the general one for any n and d is much less known. Interestingly, the distribution of Fermi particles in one d = 1) dimension has a mathematical structure similar to those found for the eigenvalues of the random matrices [3-5] and for the zeros of the Riemann zeta function [6,7], as shown below. In the following Sects. 14.2 and 14.3, explicit expressions for the pair and ternary distribution functions of the ideal Fermi gas system in any dimension are derived. We then find an expression for the n-particle distribution function as a determinant form in Sect. 14.4. Another representation for the multiparticle distribution for finite IV is given in terms of density matrix in Sect. 14.5. The explicit formula for correlation kernel which plays an essential role for the description of the multiparticle correlations in the Fermi system is derived in Sect. 14.6. The relationship with the theories for the random matrices and the Riemann zeta function is addressed in Sect. 14.7. 

Now, let us assume that Eq. 14.44 holds for the case of n-particle distribution function. The ( — l)-particle distribution function is then calculated as... 

In the preceding sections we have shown that the n-particle distribution function of ideal Fermi gas is expressed in terms of a simple determinant form (Eq. 14.44). A very analogous finding has long been known in the theory for random matrix which was initially introduced to describe the statistical distribution of nuclear energy levels [11]. Let us represent the eigenvalues of random unitary matrices U N) as exp iOj) with 1 < < and dj R. For unfolded eigenphases defined by... 

We know from statistical mechanics that a detailed understanding of the condensed noncrystalline state requires, in principle at least, the knowledge of a hierarchy of n particle distribution functions, which give die probability of finding clusters of n particles with particular positions and orientations within a system composed of N particles. It has been demonstrated in die past that it is sufficient in most cases to consider just the two lowest distribution functions, namely the singlet and the pair distribution functions. 

---