Thermal fluctuations

In the field of statistical mechanics, thermal fluctuations describe random deviations in an atomic system. These deviations occur when a system is at equilibrium, which is its steady state. Thermal fluctuations are not just side effects; they are a fundamental manifestation of temperature itself. A system at a nonzero temperature does not remain in one single microscopic state. Instead, it randomly samples all possible states. The probability of finding a system in a specific state is determined by the Boltzmann distribution.

The intensity of these fluctuations is tied directly to temperature. As a system's temperature increases, thermal fluctuations become larger and more frequent. Conversely, as the temperature approaches absolute zero, these fluctuations decrease. These movements affect many different degrees of freedom within a system. For instance, they can cause random vibrations known as phonons. They can also cause random rotations called rotons. Even electronic excitations can occur due to these random shifts.

Thermal fluctuations also impact measurable thermodynamic variables. Variables such as pressure, temperature, or entropy undergo these random wiggles. For example, a system might have a steady equilibrium pressure. However, the actual pressure will fluctuate around that equilibrium value. There are exceptions to this constant movement. The control variables of statistical ensembles do not fluctuate. These include the number of particles (N), the volume (V), and the internal energy (E) in a microcanonical ensemble.

These random forces have significant physical consequences. Thermal fluctuations act as a source of noise in many different systems. They are also responsible for both diffusion and dissipation. Dissipation includes processes like damping and viscosity. The relationship between random drift and the resistance to that drift is explained by the fluctuation-dissipation theorem. Furthermore, these fluctuations play a major role in chemical kinetics and phase transitions.

To understand these movements, scientists use the central limit theorem. In thermodynamics, a system occupies a volume of phase space. This volume is the product of configuration volume and momentum space volume. For a non-relativistic system, the energy is a quadratic form of the momenta. In systems with very high dimensionality, most of the volume lies near the surface of a hypersphere. Max Planck referred to this surface area as a "thermodynamic" probability. It represents the number of complexions compatible with a macroscopic state.

Mathematically, we can define a probability density using a partition function. This is also known as a generating function. The derivatives of the logarithm of this function generate the central moments. The first moment represents the mean energy, while the second moment represents the dispersion in energy. Because the surface area increases no faster than a power of the energy, these moments remain finite. When we expand the factor about the mean value, we often find a Gaussian distribution. This is also called a normal distribution. In this state, the average and most probable values coincide.

For systems close to equilibrium with negligible quantum effects, we can use specific expressions. If we look at a single thermodynamic variable, its probability distribution is determined by entropy. By using a Taylor expansion of entropy around its maximum, we find a Gaussian distribution. The term for the mean square fluctuation is related to the second derivative of entropy. This math allows us to predict how variables like energy or pressure will behave in small parts of a body. However, those parts must be large enough to avoid significant quantum effects.