Statistical physics

The Statistical physics deals with the description of natural phenomena, in which, although a large number of subsystems (eg particles ) is involved, but are only interested in statements of the whole or basically only incomplete information about the detailed behavior of the subsystems is available.

Importance

The Statistical physics is a fundamental physical theory whose mathematical basis sets from probability theory and the asymptotic statistics, eg form the law of large numbers, and a few physical hypotheses. With their help, be, inter alia, Derived laws of thermodynamics and justified. A branch represents the Statistical Mechanics

General

Statistical correlations can be formulated wherever an observable size of a system is dependent on the properties of its subsystems. Probability distributions come into play by that subsystems can exist in various states, but these lead to the same values ​​of the observed variables of the overall system. It is in this situation usually impractical or impossible to determine the properties of all subsystems in detail in order to infer the value of an interest to be observed size. It turns out that it is not even necessary usually to have knowledge of all the details of all subsystems in order to make viable conclusions about the overall behavior of the system.

For example, are contained in 1 liter of water as water molecules. In order to describe the flow of 1 liter of water in a pipe, it would be impractical to try to trace the paths of all 33 000 000 000 000 000 000 000 000 water molecules individually at the atomic level. It is sufficient to understand the behavior of the system at large. The Statistical physics provides concepts and methods, with those from known physical laws of nature over subsystems ( eg, particles ) statements about the system as a whole can be made.

Formulation of statistical natural laws for systems in equilibrium

In the formulation of statistical laws of nature one must first isolate the system to be described on conserved quantities. If the system has the conserved quantity E, then it is postulated that all states that are reachable without breach of this conserved quantity, equally likely to be realized ( ergodicity ). Next, we determined using physical models, the number of possible states of g as a function of this conserved quantity: g = g ( E).

If two systems S1 and S2 in interaction and facilitates the exchange of conserved quantities E1 and E2, then for the number of states of the total system S:

The overall system has a probable distribution applies when:

Because of the conservation property of E = E1 E2 = constant applies - dE1 = dE2 and

Or

Entropy

The variable s = ln g is referred to as the entropy of the system. It is up to a pre-factor ( kB is the Boltzmann constant ) is identical to the thermodynamic entropy. Subsystems si will share the conserved quantity E in contact and most often, thereby assuming those states, applies to the

Although states far outside of this equilibrium are possible, but for large systems so improbable that they can be regarded as practically impossible. The concept of entropy can thus be explained quantitatively our empirical perception that systems in contact approach a new equilibrium condition and never assume their initial states again. Entropy for example, play a major role in the fluctuation theorem.

Temperature

Consider a system of two sub-systems, in which a system is much larger than the other. The large system S will be replacing the conserved quantity E with the small system s. With sufficiently significant difference in size of the functional relationship S (E ) of the large system can be assumed to be linear, because E is changed only in small quantities. The derivative of S (E) is then a constant t.

For small differences dE the ratio of the number of neighboring states g of S (E ) is then

And for finite differences

The statistics of the small system is thus influenced by the large system, such that each state of the system is corrected with a small probability factor ~ exp (-t E). Such a statistic is called canonical ensemble. The large system is referred to as a statistical bath or reservoir. Must have the absolute values ​​of g (E ) of the reservoir in this case be no knowledge.

If considered as a concrete conserved quantity, the energy, so the small system is in thermal contact with a heat reservoir at thermodynamic temperature T

Extension to more conserved quantities

Swaps a small system in addition to a reservoir of particles N, as is the small system in contact with a particle reservoir diffuse. Again no assumptions about the absolute be g (n ) over the particle reservoir, only it is assumed that for the small area in which the system is considered small reservoir communicates with the particles:

Each state in the small system then enters with a frequency of ~ exp (-m N) exp (-t E) and the statistical ensemble described by this distribution is called the grand canonical ensemble. The size m corresponds to the factor t the chemical potential of the Teilchenreservoirs.

• Statistical Physics