Report

Statistical Mechanics Concept of the Gibbsian Ensemble In classical mechanics a state of a system is determined by knowledge of position, q, and momentum, p. p1 A microstate of a gas of N particles is specified by: 3N canonical coordinates q1, q2, …, q3N 6N-dimensional -space 3N conjugate momenta p1, p2, …, p3N or phase space A huge number of microstates correspond to the same macrostate dq1 dp1 q1 Collection of systems (mental copies) macroscopically identical but in different microstates ( p1 , q1 , ... , p 3 N , q 3 N , t ) dp1 ... p 3 N ... dq1 ... dq 3 N ( p , q , t ) d 3N pd 3N q = #of representative points at t in d3Npd3Nq probability of finding system in state with (p,q) in -space element d3Npd3Nq Another way of looking at the ensemble concept: dq1 p1 dp1 t3 t2 t1 t2 t3 t4 t5 time t4 t5 t1 q1 time trajectory spends in d3Npd3Nq probability of finding system in d3Npd3Nq Alternatively to following temporal evolution of trajectory in -space study copies 1,2,3,4,5 … at a given moment Density in -space probability density Observed value of a dynamical quantity O(p,q) dq1 p1 dp1 Ensemble average q1 Only needed when not normalized according to d 3 N p d 3 N q ( p , q ) 1 O d 3N pd d 3N 3N q O ( p, q) ( p, q, t) pd 3N q ( p, q, t) In thermal equilibrium ( p, q, t) ( p, q) O d 3N pd d 3N 3N q O ( p, q) ( p, q) pd 3N q ( p, q) The assumption O d 3N pd d 3N 3N q O ( p, q) ( p, q) pd 3N q ( p, q) lim T 1 T T O ( t ) dt 0 ergodic hypothesis Transition from classical to quantum statistics In classical mechanics a state of a system is determined by knowledge of position, q, and momentum, p. Dynamic evolution given by : trajectory in -space pi H qi , qi H pi ( p, q, t )d 3N pd 3N q =probability that a system’s phase point (p,q) is in d with ( p, q, t )d 3N pd 3N pd 3N 3N q q 1 In quantum mechanics a state of a system is determined by knowledge of the wave function q ( q ) . Thermodynamic description is given in terms of microstates that are the system’s energy eigenstates determined from H ( r 1 , r 2 , ..., r N ) E ( r 1 , r 2 , ..., r N ) Eigenfunctions labels set of quantum number Eigenenergies classical ( p, q, t )d 3N pd 3N quantum q =probability that a system’s phase point (p,q) is in d with X ( p, q, t )d 3N pd 3N pd 3N 3N 3N =probability of system being in state label by q with q 1 X ( p , q ) ( p , q , t ) d pd 3N q 1 X X Note: Later we will discuss in more detail the transition from the classical density function to the quantum mechanical density matrix