Thermal fluctuations

Atomic diffusion on the surface of a crystal. The shaking of the atoms is an example of thermal fluctuations. Likewise, thermal fluctuations provide the energy necessary for the atoms to occasionally hop from one site to a neighboring one. For simplicity, the thermal fluctuations of the blue atoms are not shown.

In statistical mechanics, thermal fluctuations are random deviations of a system from its equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they disappear altogether as temperature approaches absolute zero.

Thermal fluctuations are a basic consequence of the definition of temperature: 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, random rotations, random electronic excitations, and so forth.

Thermodynamic variables, such as pressure, temperature, or entropy, likewise undergo thermal fluctuations. For example, a system has an equilibrium pressure, but its true pressure fluctuates to some extent about the equilibrium.

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.

Distribution of fluctuations about equilibrium

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

Single variable

Suppose $x$ is a thermodynamic variable. The probability distribution $w(x)dx$ for $x$ is determined by the entropy $S$:

$w(x)\propto \exp \left(S(x)\right).$

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

$w(x)={\frac {1}{\sqrt {2\pi \langle x^{2}\rangle }}}\exp \left(-{\frac {x^{2}}{2\langle x^{2}\rangle }}\right).$

The quantity $\langle x^{2}\rangle$ is the mean square fluctuation.

Multiple variables

The above expression has a straightforward generalization to the probability distribution $w(x_{1},x_{2},\ldots ,x_{n})dx_{1}dx_{2}\ldots dx_{n}$:

$w=\prod _{i,j=1\ldots n}{\frac {1}{\left(2\pi \right)^{n/2}{\sqrt {\langle x_{i}x_{j}\rangle }}}}\exp \left(-{\frac {x_{i}x_{j}}{2\langle x_{i}x_{j}\rangle }}\right),$

where $\langle x_{i}x_{j}\rangle$ is the mean value of $x_{i}x_{j}$.

Fluctuations of the fundamental thermodynamic quantities

In the table below are given the mean square fluctuations of the thermodynamic variables $T,V,P$ and $S$ in any small part of a body. The small part must still be large enough, however, to have negligible quantum effects.

Averages $\langle x_{i}x_{j}\rangle$ of thermodyamic fluctuations. Temperature is in energy units (divide by Boltzmann's constant $k_{B}$ to get degrees). $C_{P}$ is the heat capacity at constant pressure; $C_{V}$ is the heat capacity at constant volume.

	$\Delta T$	$\Delta V$	$\Delta S$	$\Delta P$
$\Delta T$	${\frac {T^{2}}{C_{V}}}$	0	$T$	${\frac {T^{2}}{C_{V}}}\left({\frac {\partial P}{\partial T}}\right)_{V}$
$\Delta V$	0	$-T\left({\frac {\partial V}{\partial P}}\right)_{T}$	$T\left({\frac {\partial V}{\partial T}}\right)_{P}$	$-T$
$\Delta S$	$T$	$T\left({\frac {\partial V}{\partial T}}\right)_{P}$	$C_{P}$	$0$
$\Delta P$	${\frac {T^{2}}{C_{v}}}\left({\frac {\partial P}{\partial T}}\right)_{V}$	$-T$	$0$	$-T\left({\frac {\partial P}{\partial V}}\right)_{S}$

See also

• Quantum fluctuations

References

• Landau, L. D.; Lifshitz, E. M. (1985). Statistical Physics (3rd ed.). Pergamon Press. ISBN 0-08-023038-5. {{cite book}}: Unknown parameter |part= ignored (help)