In Physical statistics, the unit microcanonic are the whole of the fictitious counterparts of a real system of which energy (E), volume (V) and the number of particle (NR) are fixed. This statistical unit has important particular, because it is from this one that with the postulate of statistical physics is defined. This unit also makes it possible to determine the units canonical and large-canonical, using particle or energy exchange with a tank.

Introduction

From the point of view of the quantum Mechanical , the most complete knowledge that one can obtain from a system is the knowledge of its function of wave $\backslash \; psi\; ~\; (r\_\; 1,\; r\; \_2,\dots \; r\_N)$ which is function of the coordinates of each molecule of the system. This function of wave is solution of the equation of Schrödinger which one can write in a way condensed in the form

$H\; ~\; \backslash \; psi\; =\; E\; ~\; \backslash \; psi$
knowledge of the composition NR of the system allows to express the Hamiltonian operator $H~$, the knowledge of V specifies the boundary conditions to which must satisfy $\backslash \; psi~$. In this case, the knowledge of the energy of the system (E), eigenvalue of the equation, makes it possible to write the complete listing of all the clean functions $\backslash \; psi~$.

Assumptions

The system considered is insulated and composed of NR identical microscopic objects which can be Atome S, Molécule S, Spin S, etc…

Number of state microscopic

The full number of solution of the equation of Schrödinger is noted $\backslash \; Omega\; (E,\; V,\; NR)\; ~$. This number mathematically represents the vectorial dimension of the solutions of the equation of Schrödinger, and it depends on the variables which determines the macroscopic state system. Each microscopic state has, for a defined macroscopic state, same energy E, the same number of particles NR, and same volume V.

Postulate

The postulate of the Physique statistics specifies for an isolated system (E, V, NR fixed):

i.e. the $\backslash \; Omega~$ states corresponding to the $\backslash \; Omega~$ functions of wave of the system are also probable .

If one notes $p\_i$ the probability associated with each microphone-state I, one obtains then:

$p\_i~\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; Omega\}$

Entropy

See also: Entropy

As a whole microcanonic, the statistical entropy was defined by Boltzmann by the relation:

$\{S=k\_B\; \backslash \; cdot\; \backslash \; ln\; (\backslash \; Omega)\}$

where

• $k\_B\; =\; 1,381.10^\; \{-\; 23\}\; \backslash \; rm\; J.K^\; \{-\; 1\}$ is called the Boltzmann constant,
• $\backslash \; Omega~$ is the microscopic number of state of the system.

Measure of a size

On basis of what is known as higher, the direction of the measurement of an unspecified size $X~$ of the system has the following direction then: during the time T which measurement lasts, the system evolves/moves while passing from a microscopic state (a counterpart) to another. Any taken measurement is necessarily an average over the time of the various crossed states.

In the case of a real system, the function of wave depends on time. At any moment I, one can, to some extent, “photograph” the system in a microscopic state particular, i.e. to have a particular counterpart of it (represented by the function of wave $\backslash \; psi\_i~$, solution of the equation of Schrodïnger). However we theoretically have the list of the $\backslash \; Omega~$ allowed functions of wave and we thus know all the microscopic states by which the system is likely to pass.

In short, the function of total wave solution of the real system, is equivalent to a whole of functions of wave retorts system in each particular state which it is likely to occupy

$\backslash \; psi\; (T)\; ~\; =\; (\backslash \; psi\_1,\; \backslash \; psi\_2\; ~\dots \; ~\; \backslash \; psi\_i,\; ~\dots \; ~\; \backslash \; psi\_\; \{\backslash \; Omega\})$

Let us suppose now the measurement of the size $X~$. On basis of the postulate, and by considering a whole of counterpart of a system where each counterpart appears in the same number of specimen (for example, once each one), the value of the size $X~$ in each counterpart will be noted:

$X\_1,\; ~X\_2,\; ~\; X\_3,\; ~\dots \; ~\; X\_i,\; ~\dots \; ~\; X\_\; \{\backslash \; Omega\}\; ~$
The average of this size calculated with the whole of the counterparts is then the sum (on all the microscopic states of the system considered) of the probability of being in state I to multiply by the value $X\_i$ of this state:

$~\; =\; \backslash \; frac\; \{X\_1\; +\; X\_2\; +\; X\_3\; +\dots \; +\; X\_i\; +\dots \; +\; X\_\; \{\backslash \; Omega\}\}\; \{\backslash \; Omega\}\; =\; \backslash \; sum\_\; \{I\}\; \{X\_i\; ~\; p\_i\}$

According to the ergodic Assumption, this average must coincide with the median value measured on the real system and is defined by

$~\; =\; \backslash \; lim\_t\; ~\; \backslash \; frac\; \{1\}\; \{T\}\; \backslash \; int\_0^t\; \{X\; (T)\; dt\}$

See also: ergodic Assumption

See too

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