2016/09/24 by Massimiliano Giona, Antonio Brasiello, Silvestro Crescitelli · 16 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Binary entropy function #Chaotic #Dissipation #Entropy (arrow of time) #Ergodicity #Monotonic function #Probability density function #Quantum chaos and dynamical systems #Stochastic process #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/aa79c5
published in Journal of Physics A Mathematical and Theoretical 50(33), 335003 (Institute of Physics)
arxiv created 2016/09/24 · openalex created_date 2016/10/07 · openalex publication_date 2017/07/18 · arxiv updated 2017/08/02 · openalex updated_date 2026/08/05
Abstract In this second part, we analyze the dissipation properties of generalized Poisson–Kac (GPK) processes, considering the decay of suitable L 2 -norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo stretchy="false"></mml:mo> <mml:mspace width="thinmathspace"/> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="bold">x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mo stretchy="false"></mml:mo> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:msubsup> </mml:mstyle> </mml:math> , while the corresponding energy dissipation and entropy functions based on the overall probability density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="bold">x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> do not satisfy monotonicity requirements as a function of time. These results provide new insights on the theory of Markov operators associated with irreversible stochastic dynamics. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.