TY - JOUR

T1 - Second-order fast–slow dynamics of non-ergodic Hamiltonian systems

T2 - Thermodynamic interpretation and simulation

AU - Klar, Matthias

AU - Matthies, Karsten

AU - Reina, Celia

AU - Zimmer, Johannes

N1 - Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/12/15

Y1 - 2021/12/15

N2 - A class of fast–slow Hamiltonian systems with potential Uɛ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter ɛ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for ɛ→0 to a homogenised Hamiltonian system. We study the situation where ɛ is small but positive. First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations. Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics. Finally, we analyse for a specific fast–slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast–slow Hamiltonian system of similar accuracy.

AB - A class of fast–slow Hamiltonian systems with potential Uɛ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter ɛ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for ɛ→0 to a homogenised Hamiltonian system. We study the situation where ɛ is small but positive. First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations. Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics. Finally, we analyse for a specific fast–slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast–slow Hamiltonian system of similar accuracy.

KW - Asymptotic expansion

KW - Coarse-graining

KW - Far-from-equilibrium

KW - Many-degrees-of-freedom interaction

KW - Two-scale Hamiltonian

UR - http://www.scopus.com/inward/record.url?scp=85117865078&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2021.133036

DO - 10.1016/j.physd.2021.133036

M3 - Article

AN - SCOPUS:85117865078

SN - 0167-2789

VL - 428

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

M1 - 133036

ER -