The goal of statistical mechanics is to predict macroscopic properties on the basis of their microscopic structure
- The macroscopic properties of greatest interest to statistical mechanics are those relating to thermal equilibrium
- Hence, even though we will deal with properties that are time-dependent, they are always related to thermodynamic equilibrium
Let us consider a system composed of particles
- At each instant, the state of the system can be described by giving each particle's position ( ) and momentum ( )
- Since we are in a three-dimensional space, each particle's position and momentum is defined along each axis, thus giving us scalars for each state
- The scalars identify a point, the system's representative point ( ), within a -dimensional space, called phase space
The evolution of the system is represented by the motion of the representative point within the phase space
- The path it takes is determined by the Hamiltonian's system of differential equations of motion
- denotes the Hamiltonian of the system, which is given by
- Strictly speaking, for a system of equations of motions, there exist integrals of motions, which define a line in phase space along which the system must move
- To a very large extent, however, these integrals cannot be measured or theoratically predicted, because a small disturbance causes their values to vary greatly
In order for our argument to be consistent with thermodynamics, it is necessary that given a certain ( not very large ) number of these quantities, we can predict their values after a thermodynamic transformation
- In order to obtain a microscopic equivalent of the postulates of thermodynamics, however, we can assume that these quantities are thermodynamic observables, only this time considered as functions of the representative points
- The region in phase space in which the system is evolving can be completely identified, for thermodynamic purposes, by the values of certain observable quantities
In statistical mechanics, observables are functions defined on the phase space, that vary relatively smoothly when the representative point varies
The key to the relation between mechanical and thermodynamic properties is the microscopic expression of entropy
- It has such an important role that we call it the fundamental postulate of statistical postulate
The fundamental postulate of statistical mechanics expresses entropy as a function of the accessible volume in phase space
- The thermodynamic state of a system is defined by the values of a set of observables \
- Since these observables are functions of representative point, , the state of the system is indirectly defined by the region of phase space where the representative points all gives the same value of the set of observables \
- Since there are no changes in observables within that region, it is therefore accessible to us and we denote that region by and its volume as
- The fundamental postulate then states that
The Liouville's Theorem states that the the volume of accessible region in phase space is constant with respect to time
- As a result, the local probability density of finding a system in the neighbourhood of a representative point, , does not change when one moves along the direction guided by the Hamiltonian differential equations
- Therefore, the probability distribution does not change in time and depends only on the representative point
The phase space of positions and momenta possess a very special property with respect to time evolution
- As implied in the name "Postulate", we can not prove that entropy is indeed expressed in the equation above
- However, we can show that the properties derived from it is in agreement with that derived in thermodynamics
das