Concept# Entropy

Summary

Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication.
The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names thermodynamic function and heat-potential. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of heat to the instantan

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In this work, we are interested in elaborating models and numerical methods in order to study some phenomena which arise in the solidification of binary alloys. We propose two distinct models based on the mass and energy conservation laws as well as on classical thermodynamics. The first model is based on the irreversible process theory and therefore requires the computation of the entropy of the system to complete the description. To obtain this quantity, we develop a formalism and a method which yields numerical results. This construction is made so that the entropy function inherits some interesting mathematical properties from physical requirements. These properties allow us to elaborate and analyze mathematically an original scheme using a time discretization depending on a real parameter to solve the solidification problem. In particular, we are able to prove that the scheme is stable for all values of the time step if the parameter is chosen correctly. Furthermore, we give some numerical results to support the theory. The second model, called phase-field model, is used to describe dendrites' formation during the solidification of binary alloys. The nature of the problem constrains us to use very fine meshes in some physical regions. To reduce the number of discrete unknowns, we develop an adaptive mesh strategy based on an ad-hoc error estimator. We make numerical tests showing that the method converges on regular meshes and that the estimator is in good agreement with the true error. We also show that the mesh refinement strategy gives good results in academic and physical cases.

Daniel Maria Busiello, Deepak Gupta, Amos Maritan

A general framework to describe a vast majority of biology-inspired systems is to model them as stochastic processes in which multiple couplings are in play at the same time. Molecular motors, chemical reaction networks, catalytic enzymes, and particles exchanging heat with different baths, constitute some interesting examples of such a modelization. Moreover, they usually operate out of equilibrium, being characterized by a net production of entropy, which entails a constrained efficiency. Hitherto, in order to investigate multiple processes simultaneously driving a system, all theoretical approaches deal with them independently, at a coarse-grained level, or employing a separation of time scales. Here, we explicitly take in consideration the interplay among time scales of different processes and whether or not their own evolution eventually relaxes toward an equilibrium state in a given subspace. We propose a general framework for multiple coupling, from which the well-known formulas for the entropy production can be derived, depending on the available information about each single process. Furthermore, when one of the processes does not equilibrate in its subspace, even if much faster than all the others, it introduces a finite correction to the entropy production. We employ our framework in various simple and pedagogical examples, for which such a corrective term can be related to a typical scaling of physical quantities in play.

20202011