%0 Book Section %1 statphys23_0369 %A Ishida, H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K coarse contraction decomposed entropy equation for graining local map multibaker phase production rate space statphys23 topic-3 volume %T A decomposed equation for local entropy and entropy production in any coarse-grained system %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=369 %X The entropy production has been a central issue in non-equilibrium statistical mechanics. In order to resolve the problem coarse graining is well known to play an important role. In this work the coarse graining is carried out by the division of phase space into some partitions without restriction, such as Markov partition. Then the governing equation for probability measure is (Pauli-type) master equation. Referring the expression of master equation for ODE when a differentiable vector field is given H. Ishida and K. Momose, J. Comput. Phys., 221 (2007), 106., the equation for the local coarse-grained entropy is decomposed into six terms, i.e. (a) convection term, (b) diffusion term, (c) phase-space-volume-expansion-rate term, (d) entropy production term, and (e) a residual term in addition to unsteady term. The first two terms are antisymmetric for the reverse of a partition pair, and the spatial integration of these terms over a region reduces to its boundary integral. Therefore the integration of these terms vanishes over the region where the probability flux is zero or the periodic boundary condition is given on its boundary. We can confirm that the summation of these terms agrees with entropy flux in the multibaker map when the above-mentioned decomposition is applied to the map T. Gilbert, J. R. Dorfman and P. Gaspard, Phys. Rev. Lett., 85 (2000), 1606.. The third term (c) vanishes in Hamilton systems. However it can appear in a system upon which the condition to realize the non-equilibrium steady state is imposed. The fourth term (d) is positive definite and corresponds to the entropy production discussed in the above-mentioned multibaker map though these two do not agree. Applying the decomposition to the map, however, we can show that they coincide towards equilibrium quantitatively. The last term (e) can be decomposed into antisymetric (e1) and symmetric parts (e2). The latter behaves like a entropy source term as the term (d). However I showed that the term can be expressed by the power series of a main term, i.e. the summation of terms (c) and (d) in addition to the convection term of the master equation, and that the ratio of the spatial integration of the term (e2) over a region with multifractal structure to that of the main term goes to zero as the maximum scale of partition size approaches to zero, i.e. in the thermodynamic limit. Therefore, the term (e2) vanishes when the terms (c) and (d) remain O(1) in the limit by the multifractality. Consequently the integration of the summation of (c) and (d) terms over the whole space on which the condition for the boundary integral of probability flux to be zero, i.e. the condition for the whole probability over the region to be kept constant, is imposed vanishes in the non-equilibrium steady state in the thermodynamic limit. That is to say, the phase space volume contraction rate is shown to be equal to the entropy production in any coarse-grained system.

@incollection{statphys23_0369, abstract = {The entropy production has been a central issue in non-equilibrium statistical mechanics. In order to resolve the problem coarse graining is well known to play an important role. In this work the coarse graining is carried out by the division of phase space into some partitions without restriction, such as Markov partition. Then the governing equation for probability measure is (Pauli-type) master equation. Referring the expression of master equation for ODE when a differentiable vector field is given [H. Ishida and K. Momose, J. Comput. Phys., 221 (2007), 106.], the equation for the local coarse-grained entropy is decomposed into six terms, i.e. (a) convection term, (b) diffusion term, (c) phase-space-volume-expansion-rate term, (d) entropy production term, and (e) a residual term in addition to unsteady term. The first two terms are antisymmetric for the reverse of a partition pair, and the spatial integration of these terms over a region reduces to its boundary integral. Therefore the integration of these terms vanishes over the region where the probability flux is zero or the periodic boundary condition is given on its boundary. We can confirm that the summation of these terms agrees with entropy flux in the multibaker map when the above-mentioned decomposition is applied to the map [T. Gilbert, J. R. Dorfman and P. Gaspard, Phys. Rev. Lett., 85 (2000), 1606.]. The third term (c) vanishes in Hamilton systems. However it can appear in a system upon which the condition to realize the non-equilibrium steady state is imposed. The fourth term (d) is positive definite and corresponds to the entropy production discussed in the above-mentioned multibaker map though these two do not agree. Applying the decomposition to the map, however, we can show that they coincide towards equilibrium quantitatively. The last term (e) can be decomposed into antisymetric (e1) and symmetric parts (e2). The latter behaves like a entropy source term as the term (d). However I showed that the term can be expressed by the power series of a main term, i.e. the summation of terms (c) and (d) in addition to the convection term of the master equation, and that the ratio of the spatial integration of the term (e2) over a region with multifractal structure to that of the main term goes to zero as the maximum scale of partition size approaches to zero, i.e. in the thermodynamic limit. Therefore, the term (e2) vanishes when the terms (c) and (d) remain O(1) in the limit by the multifractality. Consequently the integration of the summation of (c) and (d) terms over the whole space on which the condition for the boundary integral of probability flux to be zero, i.e. the condition for the whole probability over the region to be kept constant, is imposed vanishes in the non-equilibrium steady state in the thermodynamic limit. That is to say, the phase space volume contraction rate is shown to be equal to the entropy production in any coarse-grained system.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Ishida, H.}, biburl = {https://www.bibsonomy.org/bibtex/22530de2f3616e39a629be3c33f10602a/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {fc71ce5a2a9445b4a1717f7ed6cd08ba}, intrahash = {2530de2f3616e39a629be3c33f10602a}, keywords = {coarse contraction decomposed entropy equation for graining local map multibaker phase production rate space statphys23 topic-3 volume}, month = {9-13 July}, timestamp = {2007-06-20T10:16:18.000+0200}, title = {A decomposed equation for local entropy and entropy production in any coarse-grained system}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=369}, year = 2007 }