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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Thermal fluctuations ： ウィキペディア英語版 Thermal fluctuations In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.〔In statistical mechanics 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. 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 (phonons), random rotations (rotons), 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. Only the 'control variables' of statistical ensembles (such as N, V and 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. == Central Limit Theorem for Thermal Fluctuations == The volume of phase space $\backslash mathcal$, occupied by a system of $2m$ degrees of freedom is the product of the configuration volume $V$ 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 $\backslash sqrt$ so that the volume of a hypersphere will vary as $\backslash sqrt^$ giving a phase volume of :$\backslash mathcal=\backslash frac,$ where $C$ is a constant depending upon the specific properties of the system and $\backslash Gamma$ is the Gamma function. In the case that this hypersphere has a very high dimensionality, $2m$, which is the usual case in thermodymamics, essentially all the volume will lie near to the surface :$\backslash Omega(E)=\backslash frac=\backslash frac,$ where we used the recursion formula $m\backslash Gamma(m)=\backslash Gamma(m+1)$. The surface area $\backslash Omega(E)$ 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 :$\backslash mathcal(\backslash beta)=\backslash int\_0^e^\backslash Omega(E)\backslash ,dE,$ which can be given a physical interpretation. The exponential decreasing factor, where $\backslash beta$ is a positive parameter, will overpower the rapidly increasing surface area so that an enormously sharp peak will develop at a certain energy $E^$. 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 :$f(E;\backslash beta)=\backslash frac(\backslash beta)\}\backslash Omega(E),$ whose integral over all energies is unity on the strength of the definition of $\backslash mathcal(\backslash beta)$, 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, :$\backslash langle\; E\backslash rangle\; =-\backslash frac,\; \backslash qquad\; \backslash \; \backslash langle(E-\backslash langle\; E\backslash rangle)^2\backslash rangle=\backslash langle(\backslash Delta\; E)^2\backslash rangle=\backslash frac,$ and so on, where the first term is the mean energy and the second one is the dispersion in energy. The fact that $\backslash Omega(E)$ increases no faster than a power of the energy ensures that these moments will be finite. 〔 〕 Therefore, we can expand the factor $e^\backslash Omega(E)$ about the mean value $\backslash langle\; E\backslash rangle$, which will coincide with $E^$ for Gaussian fluctuations (i.e. average and most probable values coincide), and retaining lowest order terms result in :$f(E;\backslash beta)=\backslash frac(\backslash beta)\}\backslash Omega(E)\backslash approx\backslash frac\}.$ 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, $f(E;\backslash beta)$, which is referred to as the canonical, or posterior, density in contrast to the prior density $\backslash Omega$, which is referred to as the 'structure' function.〔 〕 This is the central limit theorem as it applies to thermodynamic systems.〔 〕 If the phase volume increases as $E^m$, its Laplace transform, the partition function, will vary as $\backslash beta^$. Rearranging the normal distribution so that it becomes an expression for the structure function and evaluating it at $E=\backslash langle\; E\backslash rangle$ give :$\backslash Omega(\backslash langle\; E\backslash rangle)=\backslash frac(\backslash beta(\backslash langle\; E\backslash rangle))\}m\}\}$. The denominator is exactly Stirling's approximation for $m!=\backslash Gamma(m+1)$, and if the structure function retains the same functional dependency for all values of the energy, the canonical probability density, :$f(E;\backslash beta)=\backslash beta\backslash frace^$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Thermal fluctuations」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.