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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 Geometrothermodynamics ： ウィキペディア英語版 Geometrothermodynamics In physics, geometrothermodynamics (GTD) is a formalism developed recently by Hernando Quevedo to describe the properties of thermodynamic systems in terms of concepts of differential geometry. Consider a thermodynamic system in the framework of classical equilibrium thermodynamics. The states of thermodynamic equilibrium are considered as points of an abstract equilibrium space in which a Riemannian metric can be introduced in several ways. In particular, one can introduce Hessian metrics like the Fisher information metric, the Weinhold metric, the Ruppeiner metric and others, whose components are calculated as the Hessian of a particular thermodynamic potential. Another possibility is to introduce metrics which are independent of the thermodynamic potential, a property which is shared by all thermodynamic systems in classical thermodynamics. Since a change of thermodynamic potential is equivalent to a Legendre transformation, and Legendre transformations do not act in the equilibrium space, it is necessary to introduce an auxiliary space to correctly handle the Legendre transformations. This is the so-called thermodynamic phase space. If the phase space is equipped with a Legendre invariant Riemannian metric, a smooth map can be introduced that induces a thermodynamic metric in the equilibrium manifold. The thermodynamic metric can then be used with different thermodynamic potentials without changing the geometric properties of the equilibrium manifold. One expects the geometric properties of the equilibrium manifold to be related to the macroscopic physical properties. The details of this relation can be summarized in three main points: #Curvature is a measure of the thermodynamical interaction. #Curvature singularities correspond to curvature phase transitions. #Thermodynamic geodesics correspond to quasi-static processes. ==Geometric aspects== The main ingredient of GTD is a (2''n'' + 1)-dimensional manifold $\backslash mathcal$ with coordinates $Z^A=\backslash$, where $\backslash Phi$ is an arbitrary thermodynamic potential, $E^a$, $a=1,2,\backslash ldots,n$, are the extensive variables, and $I^a$ the intensive variables. It is also possible to introduce in a canonical manner the fundamental one-form $\backslash Theta\; =\; d\backslash Phi\; -\; \backslash delta\_I^a\; d\; E^b$ (summation over repeated indices) with $\backslash delta\_=(+1,\backslash ldots,+1)$, which satisfies the condition $\backslash Theta\; \backslash wedge$ (d\Theta)^n \neq 0, where $n$ is the number of thermodynamic degrees of freedom of the system, and is invariant with respect to Legendre transformations : $$ \\longrightarrow \=\\ ,\quad \Phi = \tilde \Phi - \delta_ \tilde E ^k \tilde I ^l ,\quad E^i = - \tilde I ^, \quad E^j = \tilde E ^j,\quad I^ = \tilde E ^i , \quad I^j = \tilde I ^j \ , where $i\backslash cup\; j$ is any disjoint decomposition of the set of indices $\backslash$, and $k,l=\; 1,\backslash ldots,i$. In particular, for $i=\backslash$ and $i=\backslash emptyset$ we obtain the total Legendre transformation and the identity, respectively. It is also assumed that in $\backslash mathcal$ there exists a metric $G$ which is also invariant with respect to Legendre transformations. The triad $(\backslash mathcal,\backslash Theta,G)$ defines a Riemannian contact manifold which is called the thermodynamic phase space (phase manifold). The space of thermodynamic equilibrium states (equilibrium manifold) is an n-dimensional Riemannian submanifold $\backslash mathcal\backslash subset\; \backslash mathcal$ induced by a smooth map $\backslash varphi:\backslash mathcal\backslash rightarrow\backslash mathcal$, i.e. $\backslash varphi:\backslash \; \backslash mapsto\; \backslash$, with $\backslash Phi=\backslash Phi(E^a)$ and $I^a=\; I^a(E^a)$, such that $\backslash varphi^$ *(\Theta)=\varphi^ *(d\Phi - \delta_I^dE^)=0 holds, where $\backslash varphi^$ * is the pullback of $\backslash varphi$. The manifold $\backslash mathcal$ is naturally equipped with the Riemannian metric $g=\backslash varphi^$ *(G). The purpose of GTD is to demonstrate that the geometric properties of $\backslash mathcal$ are related to the thermodynamic properties of a system with fundamental thermodynamic equation $\backslash Phi=\backslash Phi(E^a)$. The condition of invariance with respect total Legendre transformations leads to the metrics : $$ G^I = (d\Phi- \delta_I^a d E^b)^2 + \Lambda\, (\xi_ E^a I^b)\left( \delta_ dE^c d I^d\right)\ ,\quad \delta_=(1,\ldots,1) : $$ G^ = (d\Phi- \delta_I^a d E^b)^2 + \Lambda\, (\xi_ E^a I^b)\left( \eta_ dE^c d I^d\right)\ ,\quad \eta_=(-1,1,\ldots,1) where $\backslash xi\_$ is a constant diagonal matrix that can be expressed in terms of $\backslash delta\_$ and $\backslash eta\_$, and $\backslash Lambda$ is an arbitrary Legendre invariant function of $Z^A$. The metrics $G^I$ and $G^$ have been used to describe thermodynamic systems with first and second order phase transitions, respectively. The most general metric which is invariant with respect to partial Legendre transformations is : $$ G^ = (d\Phi- \delta_I^a d E^b)^2 + \Lambda\, ( E_a I_a)^ \left( dE^a d I^a\right)\ , \quad E_a= \delta_ E^b \ , \quad I_a = \delta_ I^b \ . The components of the corresponding metric for the equilibrium manifold $$ can be computed as : $$ g_ = \frac\frac G_\ . 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「Geometrothermodynamics」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.