Particle system
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Corresponding Wikipedia article: Statistical physics
Defintions
The statistical physics considers a system of large number of particles, which are moving randomly in an arbitrary spatial volume, interacting in some way with each other, with other systems and with the force fields.
The system state is a combination of the full particle energies, as the sums of their kinetic (heat, pressure) and potential (field) energies.
The equation of the steady-state particle system, that is, of the equilibrium system:
\[H=kT\,\ln\Omega=TS\tag{1}\]
\[S=k\,\ln\Omega\tag{2}\]
\(H\) is the total full energy of all the particles, or the enthalpy;
\(\Omega\) is the number of possible states of a system, which is called the statistical weight;
\(S\) is the thermodynamic entropy, which characterizes the system chaos;
\(T\) is the absolute temperature of the entire system, which characterizes the thermodynamic equilibrium;
\(k\) is the Boltzmann constant, which depends only on the units of energy and temperature.
Let an equilibrium system arbitrarily divided into the parts, or let several systems of the same temperature are brought into contact. The statistical weight (number of states) of this complicated system is obviously the product of the partial statistical weights: \[\Omega=\prod\Omega_i\tag{3}\] Equation of the complicated system as the sum of equations of the system parts: \[\sum H_i=kT\sum\ln\Omega_i=kT\ln\Omega\tag{4}\] Thus, an imbalance occurs, when the local temperature varies, or the systems with different temperatures are brought into contact. This imbalance leads to the energy redistribution and to the establishment of another common temperature, unless the imbalance is not provided by any external force.
Let the system consists of \(N\) particles, which are divided into \(M\) groups with \(N_i\) particles in each and with an average energy of a single particle \(\epsilon_i\). The system statistical weight is calculated similarly to the number of permutations and combinations: \[\Omega=N!/\prod_M\N_i!\tag{5}\] Substituting (5) into (1), and also the approximate identity for large \(N\), which is derived from the Stirling's approximation \(\lnN!\approx N(\lnN-1)\), gives: \[\sum_MN_i\frac{\epsilon_i}{kT}\approx N(\ln N-1)-\sum_MN_i(\ln N_i-1)\tag{6}\] The value of \(N\) is constant for any distribution. The increments of values \(\epsilon_i\) and \(N_i\) are related by: \[\sum_M\frac{\Delta\epsilon_i}{kT}=-\sum_M\Delta\ln N_i-1\tag{7}\] Thus, the values \(N_i\) are determined by the Gibbs distribution: \[N_i=N_0\exp\left(-\frac{\epsilon_i}{kT}\right)\tag{8}\] The value \(N_0\) does not depend on \(N_i\) and, in a case of the finite size \(M\), it can be determined by the obvious equation with a partition function \(Z\): \[\frac{N_i}{N}=\frac{\exp(-\epsilon_i/kT)}{\sum_M\exp(-\epsilon_i/kT)}=\frac{1}{Z}\exp\left(-\frac{\epsilon_i}{kT}\right)\tag{9}\]
Physical meaning of temperature
Let the total kinetic energy of the particles is: \[E=\sum_MN_i\epsilon_i=\frac{N}{Z}\sum_M\epsilon_i\exp\left(-\frac{\epsilon_i}{kT}\right)\tag{10}\] Substitution \(x=\epsilon_i/kT\) gives the equation: \[E=NkT\frac{\sum_Mxe^{-x}}{\sum_Me^{-x}}\tag{11}\] The infinitely large values \(N\) and \(M\) allow replacing a ratio of the sums (11) by a ratio of the definite integrals on the entire energy interval: \[\lim_{M\to\infty}\frac{\sum_Mxe^{-x}}{\sum_Me^{-x}}=\frac{\int_0^\infty{xe^{-x}\mathrm{d}x}}{\int_0^\infty{e^{-x}\mathrm{d}x}}=\Gamma(2)=1\tag{12}\] As a result, a large particle system with any distribution of energy has a property: \[E=NkT\tag{13}\] The expression of the average energy of a single particle defines a physical meaning of the temperature: \[\epsilon=kT\tag{14}\]
Boltzmann distribution
The continuous form of the discrete Gibbs distribution is the Boltzmann distribution: \[n=n_0\exp\left(-\frac{U}{kT}\right)\tag{15}\] For example, the gravity field energy \(U=mgh\). Substituting the atmospheric density or pressure at a zero altitude as \(n_0\), gives the well-known barometric formula: \[P(h)=P_0\exp\left(-\frac{mgh}{kT}\right)\tag{16}\]
See also
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