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 - Funding Information:
We thank Ben Leimkuhler for stimulating discussions. MK is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa) , under the project EP/L015684/1 . JZ gratefully acknowledges funding by a Royal Society Wolfson Research Merit Award . C. R. acknowledges support from NSF CAREER Award, United States , CMMI-2047506 .
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/12/15
Y1 - 2021/12/15
N2 - A class of fast–slow Hamiltonian systems with potential U
ɛ 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 U
ɛ 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
SN - 0167-2789
VL - 428
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
M1 - 133036
ER -