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Jarzynski Equality
The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two states and the irreversible work along an ensemble of trajectories joining the same states. It is named after the physicist Christopher Jarzynski (then at the University of Washington and Los Alamos National Laboratory, currently at the University of Maryland) who derived it in 1996. Fundamentally, the Jarzynski equality points to the fact that the fluctuations in the work satisfy certain constraints separately from the average value of the work that occurs in some process. Overview In thermodynamics, the free energy difference \Delta F = F_B - F_A between two states ''A'' and ''B'' is connected to the work ''W'' done on the system through the ''inequality'': : \Delta F \leq W , with equality holding only in the case of a quasistatic process, i.e. when one takes the system from ''A'' to ''B'' infinitely slowly (such that all intermediate states are in thermody ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microstate (statistical Mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. Treatments on statistical mechanics define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate. A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates. This distribution describes the probability of finding the system in a certain microstate. In the thermodynamic limit, the microstates visited by a macroscopic system during ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonequilibrium Partition Identity
The nonequilibrium partition identity (NPI) is a remarkably simple and elegant consequence of the fluctuation theorem previously known as the Kawasaki identity: : \left\langle \right\rangle = 1,\quad \forall t (Carberry et al. 2004). Thus in spite of the second law inequality which might lead one to expect that the average would decay exponentially with time, the exponential probability ratio given by the FT ''exactly'' cancels the negative exponential in the average above leading to an average which is unity for all time. The first derivation of the nonequilibrium partition identity for Hamiltonian systems was by Yamada and Kawasaki in 1967. For thermostatted deterministic systems the first derivation was by Morriss and Evans in 1985. Bibliography * * * * See also * Fluctuation theorem – Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems * Crooks fluctuation theorem – Provides a fluctuatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluctuation Theorem
The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (i.e., maximum entropy) will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously ''decrease''; the fluctuation theorem precisely quantifies this probability. Statement Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible entropy production, denoted \overline_t. The theorem states that, in systems away from equilibrium over a finite time ''t'', the ratio between the probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluctuation Theorem
The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (i.e., maximum entropy) will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously ''decrease''; the fluctuation theorem precisely quantifies this probability. Statement Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible entropy production, denoted \overline_t. The theorem states that, in systems away from equilibrium over a finite time ''t'', the ratio between the probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Overview Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol,\boldsymbol,t), also known as the Hamiltonian. The state of the system, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonequilibrium Partition Identity
The nonequilibrium partition identity (NPI) is a remarkably simple and elegant consequence of the fluctuation theorem previously known as the Kawasaki identity: : \left\langle \right\rangle = 1,\quad \forall t (Carberry et al. 2004). Thus in spite of the second law inequality which might lead one to expect that the average would decay exponentially with time, the exponential probability ratio given by the FT ''exactly'' cancels the negative exponential in the average above leading to an average which is unity for all time. The first derivation of the nonequilibrium partition identity for Hamiltonian systems was by Yamada and Kawasaki in 1967. For thermostatted deterministic systems the first derivation was by Morriss and Evans in 1985. Bibliography * * * * See also * Fluctuation theorem – Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems * Crooks fluctuation theorem – Provides a fluctuatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theorem is a proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance (to be intended in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, etc.), and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems. The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951 and expanded by Ryogo Kubo. There are antecedents to the general theorem, including Einstein's explanation of Brownian motion during his ''annus mirabilis'' and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors. Qualitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crooks Fluctuation Theorem
The Crooks fluctuation theorem (CFT), sometimes known as the Crooks equation, is an equation in statistical mechanics that relates the work done on a system during a non-equilibrium transformation to the free energy difference between the final and the initial state of the transformation. During the non-equilibrium transformation the system is at constant volume and in contact with a heat reservoir. The CFT is named after the chemist Gavin E. Crooks (then at University of California, Berkeley) who discovered it in 1998. The most general statement of the CFT relates the probability of a space-time trajectory x(t) to the time-reversal of the trajectory \tilde(t). The theorem says if the dynamics of the system satisfies microscopic reversibility, then the forward time trajectory is exponentially more likely than the reverse, given that it produces entropy, : \frac = e^. If one defines a generic reaction coordinate of the system as a function of the Cartesian coordinates of the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen's Inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations. Jensen's inequality generalizes the statement that the secant line of a convex function lies ''above'' the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for ''t'' ∈ ,1, :t f(x_1) + (1-t) f(x_2), while t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Reservoir
A thermal reservoir, also thermal energy reservoir or thermal bath, is a thermodynamic system with a heat capacity so large that the temperature of the reservoir changes relatively little when a much more significant amount of heat is added or extracted. As a conceptual simplification, it effectively functions as an infinite pool of thermal energy at a given, constant temperature. Since it can act as a source and sink of heat, it is often also referred to as a heat reservoir or heat bath. Lakes, oceans and rivers often serve as thermal reservoirs in geophysical processes, such as the weather. In atmospheric science, large air masses in the atmosphere often function as thermal reservoirs. Since the temperature of a thermal reservoir does not change during the heat transfer, the change of entropy in the reservoir is :dS_=\frac. The microcanonical partition sum Z(E) of a heat bath of temperature has the property :Z(E+\Delta E)=Z(E)e^, where k_B is the Boltzmann constant. It th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
