Statistic physics concerns itself with the description by nature phenomena their Basic Law is statistically justified. It is a physical discipline, their basis mathematical sentences (for example the law of the large numbers) and some few hypotheses form (for example the ergodic theorem) and thus very fundamentally.

Statistic laws of nature can be formulated everywhere, where an observable size of a system is statistically dependent on the characteristics of its subsystems. It is not practicable or impossible thereby to determine the characteristics of all subsystems in order to conclude from it to the value of the size which can be observed.

The nature of statistic laws of nature

Statistic laws formulate probability statements. In material scenarios of statistic physics there is however an in such a manner minted maximum for the arrival to most probable result that for all practical interests only this most probable event must be considered.

Example: one observes a Billardtisch on the 2M balls to be, which move statistically irregularly over the table. To be measured is the density of the balls. When measure for the density is counted thereby any cutout of the table observed on that the balls, the example refers to the right table half. The simplest model assumes then each ball with same probability is on the right of or on the left of the table half.

• In such a way described Model knows 22M of conditions. If each ball with equivalent probability can be on the right of or on the left of the table half, then that does not mean of these conditions from the system preferred, all conditions will become probably realized.
• There is exactly a possible distribution, with which all 2M balls is in the right table half, thus is the probability for the fact that all balls are in the right table half astronomically small and all practical interests an impossible distribution.
• The number of the possible distributions for the fact that on both table halves the same number of balls is is (2M)! /M!. The number of the possible distributions for the fact that on the observed table half two balls more than on are the other one are (2M)! /(M+1)!. Thus the relationship of the probabilities for perfect a uniform distribution and deviating from it is around a ball (M+1)! /M! = M+1. A deviation from the perfect uniform distribution is thus for large systems a rarely arising event and must not be considered.

In the described model with very many balls a law could be formulated, which postulates the uniform distribution as only observable distribution. This law is formally wrong, but for statistic statements in material scenarios practically correctly.

Formulation of statistic laws of nature

During the formulation of statistic laws of nature one must limit first the system over preservation sizes, which can be described. If the system possesses the preservation size E, then it is postulated that all conditions, which are attainable without injury of this preservation size are realized probably Next one determines the number of the possible conditions g as a function of this preservation size over physical models: g=g (E).

If one brings two systems to S1 and S2 in reciprocal effect and facilitates the exchange of the preservation sizes E1 and E2, then applies to the number of the conditions of the overall system S:

g=g_1 \, g_2

The overall system has a most probable distribution with applies:

0=g'= \ frac {D g_1} {dE_1} \, g_2+ \ frac {D g_2} {dE_2} \, g_1

Because of the preservation characteristic of E=E1+E2=konstant dE=dE1=-dE<sub>2 applies and

\ frac {1} {g_1} \ frac {D g_1} {dE} = \ frac {1} {g_2} \ frac {D g_2} {dE}

or

\ frac {D} {dE} \ LN {g_1} = \ frac {D} {dE} \ LN {g_2}

Entropy

The quantity s = LN g is called the entropy of the system. It is identical, up to a Vorfaktor (the Boltzmannkonstante KB), to the thermodynamic entropy. Subsystems SI will exchange the preservation size E so long in the contact over their contact contacts, until in pairs applies

\ frac {D s_i} {dE} = \ frac {D s_j} {dE}

The quantities SI and their functional dependence on E determine thereby completely the statistic equilibrium of the overall system. Conditions outside of this equilibrium are possible, but for sufficiently large systems so improbably that they can be regarded as practically impossible.

Temperature

A system from two subsystems is regarded, with which a system is much larger than the other one. The large system S will exchange the preservation size E with the small system s. With sufficiently clear size difference the functional connection S (E) of the large system can as linear will be accepted, since E is exchanged only in small quantities. The derivative of S (E) is a constant and as the temperature T is then designated. The large system plays the role of a statistic bath concerning the preservation size E. it with small systems an E will exchange until to the small system applies

\ frac {D s} {dE} = \ frac {1} {T}

As concrete preservation size if the energy of a system is regarded, then is identical kB/T to the thermodynamic temperature.

Distributions

The probability for observing a measured value for E in the system s in contact with a bath of the temperature t is given to the number of the conditions g (E1) and g (E2) by the comparison and results in the Boltzmann distribution:

\ Delta s = \ LN {(g_2/g_1)} = \ int_ {E_1} ^ {E_2} t \, dE = t \, (E_2-E_1) \ Rightarrow g_1/g_2 = \ exp {(- t \, \ delta E)}

In a bath thus not the measured variable E (in the statistic sense) is fixed, but the statistic distribution according to this size the Boltzmannverteilung. The statistic ensemble described by this distribution is called canonical ensemble.

Extension on more preservation sizes

If systems with two preservation sizes E and N are regarded, then the entropy S is to be formulated as function from E and N to. The large system represents then for each preservation size a bath of a certain temperature:

\ frac {\ partial S} {\ partial E} = t, \ frac {\ partial S} {\ partial N} = M.

To the small system applies in the statistic equilibrium

\ frac {\ partial s} {\ partial E} = t

and

\ frac {\ partial s} {\ partial N} = m

One keeps the connection completely similar then for the statistic ensemble in a E-bath of the temperature t and a N-bath of the temperature m

g_1/g_2 = \ exp {(- t \, \ delta E-m \, \ delta N)}

An extension on systems with more than two preservation sizes takes place similarly.

If N is the particle number of a system, then corresponds - TM the chemical potential of thermodynamics and the statistic ensemble described by this distribution is called largecanonical ensemble.

Reciprocal effects with the environment

A system s is regarded that in a E-bath with the temperature t is. The entropy of the system is to depend thereby not only on E but also on an additional parameter V, which can be given by the environment: s=s (E, V). The change of the environment parameter V will then lead to an adjustment of the statistic distribution of the quantity E in the system.

See also: Statistic mechanics, entropy

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