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In statistical mechanics, thermal fluctuations are random deviations of a 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.
Contents
- Central Limit Theorem for Thermal Fluctuations
- Distribution of fluctuations about equilibrium
- Single variable
- Multiple variables
- Fluctuations of the fundamental thermodynamic quantities
- References
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 N, V and 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 for Thermal Fluctuations
The volume of phase space
where
where we used the recursion formula
The surface area
which can be given a physical interpretation. The exponential decreasing factor, where
whose integral over all energies is unity on the strength of the definition of
and so on, where the first term is the mean energy and the second one is the dispersion in energy.
The fact that
This is the Gaussian, or normal, distribution, which is defined by its first two moments. In general, one would need all the moments to specify the probability density,
If the phase volume increases as
It follows from the expression of the first moment that
The denominator is exactly Stirling's approximation for
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.
Distribution of fluctuations about equilibrium
The expressions given below are for systems that are close to equilibrium and have negligible quantum effects.
Single variable
Suppose
If the entropy is Taylor expanded about its maximum (corresponding to the equilibrium state), the lowest order term is a Gaussian distribution:
The quantity
Multiple variables
The above expression has a straightforward generalization to the probability distribution
where
Fluctuations of the fundamental thermodynamic quantities
In the table below are given the mean square fluctuations of the thermodynamic variables