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In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium. [1] All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero.
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 the number of particules N, the volume V and the internal energy 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.
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 generate 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. [2] 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. [2] This is the central limit theorem as it applies to thermodynamic systems. [3]
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.
The expressions given below are for systems that are close to equilibrium and have negligible quantum effects. [4]
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:
The quantity is the mean square fluctuation. [4]
The above expression has a straightforward generalization to the probability distribution :
where is the mean value of . [4]
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.
In physics, the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.
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 theorem is a proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.
In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an optical driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. Propagators may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions.
In quantum field theory, the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.
The Green–Kubo relations give the exact mathematical expression for a transport coefficient in terms of the integral of the equilibrium time correlation function of the time derivative of a corresponding microscopic variable :
An ideal chain is the simplest model in polymer chemistry to describe polymers, such as nucleic acids and proteins. It assumes that the monomers in a polymer are located at the steps of a hypothetical random walker that does not remember its previous steps. By neglecting interactions among monomers, this model assumes that two monomers can occupy the same location. Although it is simple, its generality gives insight about the physics of polymers.
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution. However, they can violate the σ-additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Indeed, quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
In physics, the Maxwell–Jüttner distribution, sometimes called Jüttner–Synge distribution, is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to the Maxwell–Boltzmann distribution, the Maxwell–Jüttner distribution considers a classical ideal gas where the particles are dilute and do not significantly interact with each other. The distinction from Maxwell–Boltzmann's case is that effects of special relativity are taken into account. In the limit of low temperatures much less than , this distribution becomes identical to the Maxwell–Boltzmann distribution.
Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. Low intensity disparate light sources can be differentiated by the corresponding statistical distributions produced in the detection process.
A phonovoltaic (pV) cell converts vibrational (phonons) energy into a direct current much like the photovoltaic effect in a photovoltaic (PV) cell converts light (photon) into power. That is, it uses a p-n junction to separate the electrons and holes generated as valence electrons absorb optical phonons more energetic than the band gap, and then collects them in the metallic contacts for use in a circuit. The pV cell is an application of heat transfer physics and competes with other thermal energy harvesting devices like the thermoelectric generator.
This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry.
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