Thermal fluctuations

In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium.

[1] 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.

Thermodynamic variables, such as pressure, temperature, or entropy, likewise undergo thermal fluctuations.

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).

Thermal fluctuations play a major role in phase transitions and chemical kinetics.

degrees of freedom is the product of the configuration volume

Since the energy is a quadratic form of the momenta for a non-relativistic system, the radius of momentum space will be

is a constant depending upon the specific properties of the system and

, which is the usual case in thermodynamics, essentially all the volume will lie near to the surface where we used the recursion formula

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.

Since its integral over all energies is infinite, we might try to consider its Laplace transform which can be given a physical interpretation.

is a positive parameter, will overpower the rapidly increasing surface area so that an enormously sharp peak will develop at a certain energy

Most of the contribution to the integral will come from an immediate neighborhood about this value of the energy.

The latter name is due to the fact that the derivatives of its logarithm generate the central moments, namely, and so on, where the first term is the mean energy and the second one is the dispersion in energy.

increases no faster than a power of the energy ensures that these moments will be finite.

for Gaussian fluctuations (i.e. average and most probable values coincide), and retaining lowest order terms result in This is the Gaussian, or normal, distribution, which is defined by its first two moments.

[2] This is the central limit theorem as it applies to thermodynamic systems.

, its Laplace transform, the partition function, will vary as

Rearranging the normal distribution so that it becomes an expression for the structure function and evaluating it at

Introducing these two expressions into the expression of the structure function evaluated at the mean value of the energy leads to The denominator is exactly Stirling's approximation for

, and if the structure function retains the same functional dependency for all values of the energy, the canonical probability density, will belong to the family of exponential distributions known as gamma densities.

Consequently, the canonical probability density falls under the jurisdiction of the local law of large numbers which asserts that a sequence of independent and identically distributed random variables tends to the normal law as the sequence increases without limit.

The expressions given below are for systems that are close to equilibrium and have negligible quantum effects.

: If the entropy is Taylor expanded about its maximum (corresponding to the equilibrium state), the lowest order term is a Gaussian distribution: The quantity

[4] The above expression has a straightforward generalization to the probability distribution

[4] In the table below are given the mean square fluctuations of the thermodynamic variables

The small part must still be large enough, however, to have negligible quantum effects.