Two-dimensional Time-reversible Ergodic Maps with Provisions for Dissipation
Advanced Technology Development Center
Indian Institute of Technology Kharagpur, West Bengal, India 721302
E-mail: puneetpatra@atdc.iitkgp.ernet.in
Received:
Received: 14 November 2015; revised: 25 January 2016; accepted: 26 January 2016; published online: 01 March 2016
DOI: 10.12921/cmst.2016.22.02.001
Abstract:
A new discrete time-reversible map of a unit square onto itself is proposed. The map comprises of piecewise linear two-dimensional operations, and is able to represent the macroscopic features of both equilibrium and nonequilibrium dynamical systems. Our operations are analogous to sinusoidally driven shear in the two dimensions, and a radial compression/expansion of a point lying outside/inside a circle centred around origin. Depending upon the radius, the map transitions from being ergodic and non-dissipative (like in equilibrium situations) to a limit cycle through intermediate multifractal situations (like in nonequilibrium situations). All dissipative cases of the proposed map suggest that the Kaplan-Yorke dimension is smaller than the embedding dimension, a feature typically arising in nonequilibrium steady-states. The proposed map differs from the existing maps like the Baker map and Arnold’s cat map in the sense that (i) it is reversible, and (ii) it generates an intricate multifractal phase-space portrait.
Key words:
dissipative maps, ergodic maps, Lyapunov exponents, time reversibility
References:
[1] Wm.G. Hoover and C.G. Hoover, Time reversibility, computer
simulation, algorithms, chaos, World Scientific, 2012.
[2] S. Nosé, A unified formulation of the constant temperature
molecular dynamics methods, The Journal of Chemical
Physics 81(1), 511-519 (1984).
[3] Wm.G. Hoover, Canonical dynamics: Equilibrium phasespace
distributions, Physical Review A 31(3), 1695-1697
(1985).
[4] Wm.G. Hoover and B.L. Holian, Kinetic moments method
for the canonical ensemble distribution, Physics Letters A
211(5), 253-257 (1996).
[5] G.J. Martyna, M.L. Klein, and M. Tuckerman, Nosé-Hoover
chains: The canonical ensemble via continuous dynamics,
The Journal of Chemical Physics 97(4), 2635-2643 (1992).
[6] P. K. Patra and B. Bhattacharya, A deterministic thermostat
for controlling temperature using all degrees of freedom, The
Journal of Chemical Physics 140(6), 064106 (2014).
[7] P.K. Patra and B. Bhattacharya, An ergodic configurational
thermostat using selective control of higher order temperatures,
The Journal of Chemical Physics 142(19), 194103
(2015).
[8] Wm.G. Hoover, O. Kum, and H.A. Posch, Time-reversible
dissipative ergodic maps, Physical Review E 53(3), 2123
(1996).
[9] D.J. Evans and G. Morriss, Statistical mechanics of nonequilibrium
liquids, Cambridge University Press, 2008.
[10] P.K. Patra and B. Bhattacharya, Nonergodicity of the Nose-
Hoover chain thermostat in computationally achievable time,
Physical Review E 90(4), 043304 (2014).
[11] P.K. Patra, J.C. Sprott, Wm.G. Hoover, and C.G. Hoover,
Deterministic time-reversible thermostats: chaos, ergodicity,
and the zeroth law of thermodynamics, Molecular Physics
113(17-18), 2863-2872 (2015).
[12] H.A. Posch and Wm.G. Hoover, Time-reversible dissipative
attractors in three and four phase-space dimensions, Physical
Review E 55(6), 6803 (1997).
[13] P. Grassberger and I. Procaccia, Measuring the strangeness
of strange attractors, In The Theory of Chaotic Attractors,
pages 170-189 Springer, 2004.
[14] Wm.G. Hoover and C.G. Hoover, Time-symmetry breaking
in Hamiltonian mechanics, arXiv preprint arXiv:1302.2533
(2013).
[15] Wm.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-
Reversible Chaos for Gibbs’ Canonical Oscillator, arXiv
preprint arXiv:1503.06749 (2015).
[16] O.B. Isaeva, A.Y. Jalnine, and S.P. Kuznetsov, Arnold’s cat
map dynamics in a system of coupled nonautonomous van der
Pol oscillators, Phys. Rev. E, 74, 046207 (2006).
[17] Wm.G. Hoover and C.G. Hoover, Simulation and Control of
Chaotic Nonequilibrium Systems, Advanced series in nonlinear
dynamics World Scientific, 2015.
[18] L. Rondoni and G.P. Morriss, Stationary nonequilibrium
ensembles for thermostated systems, Phys. Rev. E, 53, 2143-
2153 (1996).
[19] Wm.G. Hoover and C.G. Hoover, Time-Reversible Ergodic
Maps and the 2015 Ian Snook Prize, arXiv preprint
arXiv:1507.01645 (2015).
[20] T. Riley, A. Goucher, Beautiful Testing: Leading Professionals
Reveal How They Improve Software, O’Reilly Media, Inc.,
2009.
[21] G. Marsaglia, DIEHARD: a battery of tests of randomness,
See http://stat.fsu.edu/~geo/diehard.html (1996).
A new discrete time-reversible map of a unit square onto itself is proposed. The map comprises of piecewise linear two-dimensional operations, and is able to represent the macroscopic features of both equilibrium and nonequilibrium dynamical systems. Our operations are analogous to sinusoidally driven shear in the two dimensions, and a radial compression/expansion of a point lying outside/inside a circle centred around origin. Depending upon the radius, the map transitions from being ergodic and non-dissipative (like in equilibrium situations) to a limit cycle through intermediate multifractal situations (like in nonequilibrium situations). All dissipative cases of the proposed map suggest that the Kaplan-Yorke dimension is smaller than the embedding dimension, a feature typically arising in nonequilibrium steady-states. The proposed map differs from the existing maps like the Baker map and Arnold’s cat map in the sense that (i) it is reversible, and (ii) it generates an intricate multifractal phase-space portrait.
Key words:
dissipative maps, ergodic maps, Lyapunov exponents, time reversibility
References:
[1] Wm.G. Hoover and C.G. Hoover, Time reversibility, computer
simulation, algorithms, chaos, World Scientific, 2012.
[2] S. Nosé, A unified formulation of the constant temperature
molecular dynamics methods, The Journal of Chemical
Physics 81(1), 511-519 (1984).
[3] Wm.G. Hoover, Canonical dynamics: Equilibrium phasespace
distributions, Physical Review A 31(3), 1695-1697
(1985).
[4] Wm.G. Hoover and B.L. Holian, Kinetic moments method
for the canonical ensemble distribution, Physics Letters A
211(5), 253-257 (1996).
[5] G.J. Martyna, M.L. Klein, and M. Tuckerman, Nosé-Hoover
chains: The canonical ensemble via continuous dynamics,
The Journal of Chemical Physics 97(4), 2635-2643 (1992).
[6] P. K. Patra and B. Bhattacharya, A deterministic thermostat
for controlling temperature using all degrees of freedom, The
Journal of Chemical Physics 140(6), 064106 (2014).
[7] P.K. Patra and B. Bhattacharya, An ergodic configurational
thermostat using selective control of higher order temperatures,
The Journal of Chemical Physics 142(19), 194103
(2015).
[8] Wm.G. Hoover, O. Kum, and H.A. Posch, Time-reversible
dissipative ergodic maps, Physical Review E 53(3), 2123
(1996).
[9] D.J. Evans and G. Morriss, Statistical mechanics of nonequilibrium
liquids, Cambridge University Press, 2008.
[10] P.K. Patra and B. Bhattacharya, Nonergodicity of the Nose-
Hoover chain thermostat in computationally achievable time,
Physical Review E 90(4), 043304 (2014).
[11] P.K. Patra, J.C. Sprott, Wm.G. Hoover, and C.G. Hoover,
Deterministic time-reversible thermostats: chaos, ergodicity,
and the zeroth law of thermodynamics, Molecular Physics
113(17-18), 2863-2872 (2015).
[12] H.A. Posch and Wm.G. Hoover, Time-reversible dissipative
attractors in three and four phase-space dimensions, Physical
Review E 55(6), 6803 (1997).
[13] P. Grassberger and I. Procaccia, Measuring the strangeness
of strange attractors, In The Theory of Chaotic Attractors,
pages 170-189 Springer, 2004.
[14] Wm.G. Hoover and C.G. Hoover, Time-symmetry breaking
in Hamiltonian mechanics, arXiv preprint arXiv:1302.2533
(2013).
[15] Wm.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-
Reversible Chaos for Gibbs’ Canonical Oscillator, arXiv
preprint arXiv:1503.06749 (2015).
[16] O.B. Isaeva, A.Y. Jalnine, and S.P. Kuznetsov, Arnold’s cat
map dynamics in a system of coupled nonautonomous van der
Pol oscillators, Phys. Rev. E, 74, 046207 (2006).
[17] Wm.G. Hoover and C.G. Hoover, Simulation and Control of
Chaotic Nonequilibrium Systems, Advanced series in nonlinear
dynamics World Scientific, 2015.
[18] L. Rondoni and G.P. Morriss, Stationary nonequilibrium
ensembles for thermostated systems, Phys. Rev. E, 53, 2143-
2153 (1996).
[19] Wm.G. Hoover and C.G. Hoover, Time-Reversible Ergodic
Maps and the 2015 Ian Snook Prize, arXiv preprint
arXiv:1507.01645 (2015).
[20] T. Riley, A. Goucher, Beautiful Testing: Leading Professionals
Reveal How They Improve Software, O’Reilly Media, Inc.,
2009.
[21] G. Marsaglia, DIEHARD: a battery of tests of randomness,
See http://stat.fsu.edu/~geo/diehard.html (1996).
