The Anomalous Thermal Conductivity of Quasi-one-dimensional Hard Disks
School of Physics, University of New South Wales
Sydney NSW 2052, Australia
E-mail: g.morriss@unsw.edu.au
Received:
Received: 20 December 2016; revised: 10 March 2017; accepted: 15 March 2017; published online: 15 April 2017
DOI: 10.12921/cmst.2016.00000065
Abstract:
We confirm that the conduction of heat in a system of quasi-one-dimension hard disks, with mechanically connected heat reservoirs of different temperatures, is anomalous. We consider systems of different sizes at the same density with the same externally applied temperature gradient and observe that the anomalous behaviour changes with system size. For systems with less than 1000 disks we find that the heat flux vector varies with the square root of the number of disks whereas for systems with more than 1000 disks the heat flux vector varies with the 2/3 power of the number of disks.
Key words:
References:
[1] S. Lepri, R. Livi, A. Politi, Thermal conduction in classical low-dimensional lattices, Phys. Rep. 377, 1 (2003).
[2] A. Dhar, Heat conduction in the disordered harmonic chain revisited, Phys. Rev. Lett. 86, 5882 (2001).
[3] S. Lepri, R. Livi, A. Politi, Heat conduction in chains of nonlinear oscillators, Phys. Rev. Lett. 78, 1896 (1997).
[4] S. Lepri, R. Livi, A. Politi, On the anomalous thermal conductivity of one-dimensional lattices, Europhys. Lett. 43, 271 (1998).
[5] T. Hatano, Heat conduction in the diatomic Toda lattice revisited, Phys. Rev. E 59, R1 (1999).
[6] P. Grassberger, W. Nadler, L. Yang, Heat conduction and entropy production in a one-dimensional hard-particle gas, Phys. Rev. Lett. 89, 180601 (2002).
[7] J.M. Deutsch, O. Narayan, One-dimensional heat conductivity exponent from a random collision model Phys. Rev. E 68, 010201 (2003).
[8] J.M. Deutsch, O. Narayan, Correlations and scaling in one-dimensional heat conduction Phys. Rev. E 68, 041203 (2003).
[9] Ya.G. Sinai, N.I. Chernov, Ergodic properties of certain systems of two-dimensional disks and 3-dimensional balls, Russ. Math. Surveys 42, 181 (1987).
[10] N. Simányi, D. Szász, The k-property of hamiltonian systems with restricted hard ball interactions Math. Res. Lett. 2, 751 (1995).
[11] T. Prosen, D.K. Campbell, Normal and anomalous heat transport in one-dimensional classical lattices, CHAOS 15, 015117 (2005).
[12] J-S Wang, B. Li, Mode-coupling theory and molecular dynamics simulation for heat conduction in a chain with transverse motions, Phys. Rev. E 70, 021204 (2004).
[13] L. Delfini, S. Lepri, R. Livi, A. Politi, Self-consistent mode-coupling approach to one-dimensional heat transport, Phys. Rev. E 73, 060201(R) (2006).
[14] O. Narayan, S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002).
[15] L. Delfini, S. Lepri, R. Livi and A. Politi, Anomalous kinetics and transport from 1D self-consistent mode-coupling theory, J. Stat. Mech. 7, P02007 (2007).
[16] H. van Beijeren, Exact Results for Anomalous Transport in One-Dimensional Hamiltonian Systems, Phys. Rev. Lett. 108 180601 (2012).
[17] M. Prähofer, H. Spohn, Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys. 115, 255 (2004).
[18] T. Sasamoto, H. Spohn, Superdiffusivity of the 1D Lattice Kardar-Parisi-Zhang Equation, J. Stat. Phys. 137, 917 (2009).
[19] T. Taniguchi, G.P. Morriss, Lyapunov Modes for a Nonequilibrium System with a Heat Flux, Comptes Rendus Physique, 8, 625 (2007).
[20] J.A. McLennan, Introduction to Non-equilibrium Statistical Mechanics, Prentice-Hall, Englewood Cliffs NJ, 1989.
[21] D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd Edition, Cambridge University Press, Cambridge, 2008.
[22] G.P. Morriss, D. Truant, Deterministic Thermal Reservoirs, Entropy, 14, 1011 (2012).
[23] G.P. Morriss, D. Truant, Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems, Phys. Rev. E, 87, 062144 (2013).
[24] G.P. Morriss, D.P. Truant, A Review of the Hydrodynamic Lyapunov Modes of Hard Disk Systems, J. Phys. A. 46 254010 (2013).
[25] T. Chung, D. Truant, G.P. Morriss, Lyapunov Modes as Fields, Phys. Rev. E 83, 046216 (2011).
[26] D.P. Truant, G.P. Morriss, Backward and covariant Lyapunov vectors and exponents for hard disk systems with a steady heat current, Phys. Rev. E 90, 052907 (2014).
[27] G.P. Morriss, Calculating the Local Nonequilibrium Configurational Entropy in Quasi-one-dimensional Heat Conduction, Molecular Simulations 42(6-7) 474–483 (2016).
We confirm that the conduction of heat in a system of quasi-one-dimension hard disks, with mechanically connected heat reservoirs of different temperatures, is anomalous. We consider systems of different sizes at the same density with the same externally applied temperature gradient and observe that the anomalous behaviour changes with system size. For systems with less than 1000 disks we find that the heat flux vector varies with the square root of the number of disks whereas for systems with more than 1000 disks the heat flux vector varies with the 2/3 power of the number of disks.
Key words:
References:
[1] S. Lepri, R. Livi, A. Politi, Thermal conduction in classical low-dimensional lattices, Phys. Rep. 377, 1 (2003).
[2] A. Dhar, Heat conduction in the disordered harmonic chain revisited, Phys. Rev. Lett. 86, 5882 (2001).
[3] S. Lepri, R. Livi, A. Politi, Heat conduction in chains of nonlinear oscillators, Phys. Rev. Lett. 78, 1896 (1997).
[4] S. Lepri, R. Livi, A. Politi, On the anomalous thermal conductivity of one-dimensional lattices, Europhys. Lett. 43, 271 (1998).
[5] T. Hatano, Heat conduction in the diatomic Toda lattice revisited, Phys. Rev. E 59, R1 (1999).
[6] P. Grassberger, W. Nadler, L. Yang, Heat conduction and entropy production in a one-dimensional hard-particle gas, Phys. Rev. Lett. 89, 180601 (2002).
[7] J.M. Deutsch, O. Narayan, One-dimensional heat conductivity exponent from a random collision model Phys. Rev. E 68, 010201 (2003).
[8] J.M. Deutsch, O. Narayan, Correlations and scaling in one-dimensional heat conduction Phys. Rev. E 68, 041203 (2003).
[9] Ya.G. Sinai, N.I. Chernov, Ergodic properties of certain systems of two-dimensional disks and 3-dimensional balls, Russ. Math. Surveys 42, 181 (1987).
[10] N. Simányi, D. Szász, The k-property of hamiltonian systems with restricted hard ball interactions Math. Res. Lett. 2, 751 (1995).
[11] T. Prosen, D.K. Campbell, Normal and anomalous heat transport in one-dimensional classical lattices, CHAOS 15, 015117 (2005).
[12] J-S Wang, B. Li, Mode-coupling theory and molecular dynamics simulation for heat conduction in a chain with transverse motions, Phys. Rev. E 70, 021204 (2004).
[13] L. Delfini, S. Lepri, R. Livi, A. Politi, Self-consistent mode-coupling approach to one-dimensional heat transport, Phys. Rev. E 73, 060201(R) (2006).
[14] O. Narayan, S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002).
[15] L. Delfini, S. Lepri, R. Livi and A. Politi, Anomalous kinetics and transport from 1D self-consistent mode-coupling theory, J. Stat. Mech. 7, P02007 (2007).
[16] H. van Beijeren, Exact Results for Anomalous Transport in One-Dimensional Hamiltonian Systems, Phys. Rev. Lett. 108 180601 (2012).
[17] M. Prähofer, H. Spohn, Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys. 115, 255 (2004).
[18] T. Sasamoto, H. Spohn, Superdiffusivity of the 1D Lattice Kardar-Parisi-Zhang Equation, J. Stat. Phys. 137, 917 (2009).
[19] T. Taniguchi, G.P. Morriss, Lyapunov Modes for a Nonequilibrium System with a Heat Flux, Comptes Rendus Physique, 8, 625 (2007).
[20] J.A. McLennan, Introduction to Non-equilibrium Statistical Mechanics, Prentice-Hall, Englewood Cliffs NJ, 1989.
[21] D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd Edition, Cambridge University Press, Cambridge, 2008.
[22] G.P. Morriss, D. Truant, Deterministic Thermal Reservoirs, Entropy, 14, 1011 (2012).
[23] G.P. Morriss, D. Truant, Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems, Phys. Rev. E, 87, 062144 (2013).
[24] G.P. Morriss, D.P. Truant, A Review of the Hydrodynamic Lyapunov Modes of Hard Disk Systems, J. Phys. A. 46 254010 (2013).
[25] T. Chung, D. Truant, G.P. Morriss, Lyapunov Modes as Fields, Phys. Rev. E 83, 046216 (2011).
[26] D.P. Truant, G.P. Morriss, Backward and covariant Lyapunov vectors and exponents for hard disk systems with a steady heat current, Phys. Rev. E 90, 052907 (2014).
[27] G.P. Morriss, Calculating the Local Nonequilibrium Configurational Entropy in Quasi-one-dimensional Heat Conduction, Molecular Simulations 42(6-7) 474–483 (2016).
