Thermal fluctuations

Thermal fluctuations are small, random deviations in a thermodynamic variable. Arising from sources such as collisions of molecules and lattice vibrations, they are a source of noise in many systems. They are both a source of diffusion and of 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 $S$:

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

Expanding the entropy about its maximum (corresponding to the equilibrium state) gives

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

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

Multiple variables

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

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

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

Fluctuations of the fundamental thermodynamic quantities

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

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)