In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero. Thermal fluctuations are a basic manifestation of the temperature of systems: A system at nonzero temperature does not stay in its equilibrium microscopic state, but instead randomly samples all possible states, with probabilities given by the Boltzmann distribution. Thermal fluctuations generally affect all the degrees of freedom of a system: There can be random vibrations (phonons), random rotations (rotons), random electronic excitations, and so forth. Thermodynamic variables, such as pressure, temperature, or entropy, likewise undergo thermal fluctuations. For example, for a system that has an equilibrium pressure, the system pressure fluctuates to some extent about the equilibrium value. Only the 'control variables' of statistical ensembles (such as the number of particules N, the volume V and the internal energy E in the microcanonical ensemble) do not fluctuate. Thermal fluctuations are a source of noise in many systems. The random forces that give rise to thermal fluctuations are a source of both diffusion and dissipation (including damping and viscosity). The competing effects of random drift and resistance to drift are related by the fluctuation-dissipation theorem. Thermal fluctuations play a major role in phase transitions and chemical kinetics.
Central limit theorem The volume of phase space V {\displaystyle {\mathcal {V}}} , occupied by a system of 2 m {\displaystyle 2m} degrees of freedom is the product of the configuration volume V {\displaystyle V} and the momentum space volume. Since the energy is a quadratic form of the momenta for a non-relativistic system, the radius of momentum space will be E {\displaystyle {\sqrt {E}}} so that the volume of a hypersphere will vary as E 2 m {\displaystyle {\sqrt {E}}^{2m}} giving a phase volume of
V = ( C ⋅ E ) m Γ ( m + 1 ) , {\displaystyle {\mathcal {V}}={\frac {(C\cdot E)^{m}}{\Gamma (m+1)}},}
where C {\displaystyle C} is a constant depending upon the specific properties of the system and Γ {\displaystyle \Gamma } is the Gamma function. In the case that this hypersphere has a very high dimensionality, 2 m {\displaystyle 2m} , which is the usual case in thermodynamics, essentially all the volume will lie near to the surface
Ω ( E ) = ∂ V ∂ E = C m ⋅ E m − 1 Γ ( m ) , {\displaystyle \Omega (E)={\frac {\partial {\mathcal {V}}}{\partial E}}={\frac {C^{m}\cdot E^{m-1}}{\Gamma (m)}},}
where we used the recursion formula m Γ ( m ) = Γ ( m + 1 ) {\displaystyle m\Gamma (m)=\Gamma (m+1)} . The surface area Ω ( E ) {\displaystyle \Omega (E)} has its legs in two worlds: (i) the macroscopic one in which it is considered a function of the energy, and the other extensive variables, like the volume, that have been held constant in the differentiation of the phase volume, and (ii) the microscopic world where it represents the number of complexions that is compatible with a given macroscopic state. It is this quantity that Planck referred to as a 'thermodynamic' probability. It differs from a classical probability inasmuch as it cannot be normalized; that is, its integral over all energies diverges—but it diverges as a power of the energy and not faster. Since its integral over all energies is infinite, we might try to consider its Laplace transform
Z ( β ) = ∫ 0 ∞ e − β E Ω ( E ) d E , {\displaystyle {\mathcal {Z}}(\beta )=\int _{0}^{\infty }e^{-\beta E}\Omega (E)\,dE,}
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