In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.
[In ]statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
they are often simply referred to as fluctuations. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches
absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
.
Thermal fluctuations are a basic manifestation of the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
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
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
.
Thermal fluctuations generally affect all the
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of a system: There can be random vibrations (
phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
s), random rotations (
roton
In theoretical physics, a roton is an elementary excitation, or quasiparticle, seen in superfluid helium-4 and Bose–Einstein condensates with long-range dipolar interactions or spin-orbit coupling. The dispersion relation of elementary excita ...
s), random electronic excitations, and so forth.
Thermodynamic variables, such as pressure, temperature, or
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, 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
In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it cannot ...
) do not fluctuate.
Thermal fluctuations are a source of
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
in many systems. The random forces that give rise to thermal fluctuations are a source of both
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
and
dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
(including
damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
and
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
). The competing effects of random drift and resistance to drift are related by the
fluctuation-dissipation theorem
The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the the ...
. Thermal fluctuations play a major role in
phase transitions
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...
and
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
.
Central limit theorem
The volume of phase space
, occupied by a system of
degrees of freedom is the product of the configuration volume
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
so that the volume of a hypersphere will vary as
giving a phase volume of
:
where
is a constant depending upon the specific properties of the system and
is the Gamma function. In the case that this hypersphere has a very high dimensionality,
, which is the usual case in thermodynamics, essentially all the volume will lie near to the surface
:
where we used the recursion formula
.
The surface area
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
:
which can be given a physical interpretation. The exponential decreasing factor, where
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. This enables the definition of a proper probability density according to
:
whose integral over all energies is unity on the strength of the definition of
, which is referred to as the partition function, or generating function. The latter name is due to the fact that the derivatives of its logarithm generates the central moments, namely,
:
and so on, where the first term is the mean energy and the second one is the dispersion in energy.
The fact that
increases no faster than a power of the energy ensures that these moments will be finite.
Therefore, we can expand the factor
about the mean value
, which will coincide with
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. In general, one would need all the moments to specify the probability density,
, which is referred to as the canonical, or posterior, density in contrast to the prior density
, which is referred to as the 'structure' function.
This is the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
as it applies to thermodynamic systems.
If the phase volume increases as
, 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
give
:
It follows from the expression of the first moment that
, while from the second central moment,
. 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.
Distribution about equilibrium
The expressions given below are for systems that are close to equilibrium and have negligible quantum effects.
Single variable
Suppose
is a thermodynamic variable. The probability distribution
for
is determined by the entropy
:
:
If the entropy is
Taylor expanded about its maximum (corresponding to the
equilibrium state), the lowest order term is a
Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
:
:
The quantity
is the mean square fluctuation.
Multiple variables
The above expression has a straightforward generalization to the probability distribution
:
:
where
is the mean value of
.
[
]
Fluctuations of the fundamental thermodynamic quantities
In the table below are given the mean square fluctuations of the thermodynamic variables and in any small part of a body. The small part must still be large enough, however, to have negligible quantum effects.
See also
*Quantum fluctuation
In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s
Notes
References
*
*
*
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Statistical mechanics