9/11/2023 0 Comments Entropy of mixingWe then focus on fluid mixing as a case study which best illustrates our approach and which is an important and well studied problem in the biophysics of lipid membranes. Since the entropy is also defined as the volume of phase space available to the system, correlations always lead to a decrease of entropy as they are equivalent to constraints on the configurations of the system which reduce the volume of the phase space.Īfter describing the problem of quantifying disorder in multicomponent systems, we introduce our new method based on the conditional entropy, and we validate it by computing the entropy in the Ising model. However, it highly overestimates the value of the entropy close to the critical temperature, as the correlation length diverges to infinity at continuous phase transitions. For example, this is a good approximation for the Ising ferromagnet in the high temperature regime, when the correlations between neighbouring spins ( or ) are small. when there are no correlations between different particles. This is a drastic simplification, which is only accurate when the -point joint probability distribution can be factorised into the single particle probabilities, i.e. (2)where is the probability of the single particle state. The simplest estimate of the Shannon entropy is given by the mean field approximation. Its main advantage is the capturing of local correlations between particles or states, which are an important factor for characterising disorder-order transitions. Our approach employs the concept of conditional entropy, which derives from the measure of entropy in images and complex networks. It can be easily implemented for use on physical simulation and experimental datasets. Here we present a widely applicable method to quantify disorder in systems at or away from equilibrium, based on Shannon and conditional entropy. In previous work, the Shannon entropy has been used successfully to quantify the order in fluid mixtures. Hence, it can be a useful tool to describe any macro-state of the system. The information-theoretical derivation of entropy is not restricted to thermodynamic equilibrium and it can be computed directly from the observed frequencies of configurations. (1)where the sum is performed over all possible configurations of the system, and represents the frequency of occurrence of the N-particle state. We here use the Shannon entropy from information theory, which is defined as. In order to develop a useful, more general measure of system disorder, it is therefore necessary to consider alternative approaches. However, since experiments or simulations are often monitoring systems away from equilibrium, and the system Hamiltonian is often unknown in experiments, a simple formulation using the thermodynamic entropy is not readily available. Because of the widespread occurrence of these phenomena, it is desirable to obtain methods to quantify the local and global level of disorder in a system, which can be generally applied to a broad range of systems.Īt equilibrium, disorder can be quantified by the thermodynamic entropy, which typically necessitates the explicit knowledge of the partition function of the system. They play a major role in the description of the behaviour of liquids and solids, the level of spin alignment in ferromagnetic systems, and domain formation in biological fluids such as membranes. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.įunding: MS and UZ acknowledge support from the Scottish Universities' Physics Alliance ( MS, CM and UZ acknowledge external funding from the UK National Physical Laboratory ( The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist.ĭisorder-order Transitions and the Shannon Entropyĭisorder-order transitions are important physical phenomena that are commonly addressed both by simulations and experiments. Received: FebruAccepted: ApPublished: June 10, 2013Ĭopyright: © 2013 Brandani et al. PLoS ONE 8(6):Įditor: Christof Markus Aegerter, University of Zurich, Switzerland Citation: Brandani GB, Schor M, MacPhee CE, Grubmüller H, Zachariae U, Marenduzzo D (2013) Quantifying Disorder through Conditional Entropy: An Application to Fluid Mixing.
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