Abstract
Low dimensional nonequilibrium steady-state (NESS) systems are essential in understanding the microscopic origins of macroscopic laws. A one-dimensional chain, thermostatted either end to a different temperature, allows the study of heat flow in small scale systems. It is essential in understanding the origins of Fourier’s laws. In recent decades, many such systems have been studied in detail. The model of alternating masses connected through harmonic springs is one of them. With temperature gradient enforced and the chain in steady state, the particles’ kinetic temperatures take two values alternatingly. The higher massed particles can have lower or higher temperatures depending on whether there are even or odd numbers of particles in the chain. This model can provide us with insights into the heat flow in many nanomaterials, especially those with different masses along their lengths. The objective of this study is to understand how the temperature profile and the thermal conductivity change when the particles of a chain have a finite size and can collide with each other. Unlike the pedagogical models studied previously, our model is more realistic as two adjacent particles can never cross each other. Incorporating collisions reduce the alternating temperatures of the particles. Furthermore, collisions make the temperature profile more linear and reduce the boundary jumps at the same time. We have also computed the conductivity of the different chains by time-averaging the heat current. The thermal conductivity of the modified harmonic chains approaches length independence asymptotically, indicating a substantial modal energy redistribution in the chains with collisions, which is unlike that observed when the collisions are absent.
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References
Fujii M, Zhang X, Xie H, Ago H, Takahashi K, Ikuta T, Abe H, Shimizu T (2005) Measuring the thermal conductivity of a single carbon nanotube. Phys Rev Lett 95:065502
Lepri S, Livi R, Politi A (2003) Thermal conduction in classical low-dimensional lattices. Phys Rep 377:1–80
Lepri S, Livi R, Politi A (1997) Heat conduction in chains of nonlinear oscillators. Phys Rev Lett 78:1896
Chen D, Aubry S, Tsironis G (1996) Breather mobility in discrete φ4 nonlinear lattices. Phys Rev Lett 77:4776
Casati G, Ford J, Vivaldi F, Visscher WM (1984) One-dimensional classical many-body system having a normal thermal conductivity. Phys Rev Lett 52:1861
Giardina C, Kurchan J (2005) The fourier law in a momentum-conserving chain. J Stat Mech Theory Exp 3422005:P05009
Giardina C, Livi R, Politi A, Vassalli M (2000) Finite thermal conductivity in 1d lattices. Phys Rev Lett 84:2144
Lee-Dadswell G, Turner E, Ettinger J, Moy M (2010) Momentum conserving one-dimensional system with a finite thermal conductivity. Phys Rev E 82:061118
Prosen T, Campbell DK (2000) Momentum conservation implies anomalous energy transport in 1d classical lattices. Phys Rev Lett 38(84):2857
Kannan V, Dhar A, Lebowitz JL (2012, April 12) Nonequilibrium stationary state of a harmonic crystal with alternating masses. Phys Rev E 85(4):041118
Mai T, Dhar A, Narayan O (2007, May 4) Equilibration and universal heat conduction in fermi-pasta-ulam chains. Phys Rev Lett 98(18):184301
Bruggeman R, Verhulst F (2019, February 15) Near-integrability and recurrence in FPU chains with alternating masses. J Nonlinear Sci 29(1):183–206
Bhattacharyya S, Patra PK (2020, May 20) Thermal transport properties of one-dimensional Φ4 chains with colliding particles. Commun Nonlinear Sci Numer Simul 105323
Bhattacharyya S, Patra PK (2020 June 23) Thermal transport characteristics of Fermi-Pasta-Ulam chains undergoing soft-sphere type collisions. arXiv preprint arXiv:2006.13238
Martyna GJ, Klein ML, Tuckerman M (1992) Nosé–hoover chains: the canonical ensemble via continuous dynamics. J Chem Phys 97:2635–2643
Nosé S (1984) A unified formulation of the constant temperature molecular dynamics methods. J Chem Phys 81:511–519
Patra P, Bhattacharya B (2014) A deterministic thermostat for controlling temperature using all degrees of freedom. J Chem Phys 140:064106
Patra PK, Sprott JC, Hoover WG, Hoover CG (2015) Deterministic time-reversible thermostats: chaos, ergodicity, and the zeroth law of thermodynamics. Mol Phys 113:2863–2872
Patra PK, Bhattacharya B (2015, May 21) An ergodic configurational thermostat using selective control of higher order temperatures. J Chem Phys 142(19):194103
Davidchack RL, Handel R, Tretyakov MV (2009, June 21) Langevin thermostat for rigid body dynamics. J Chem Phys 130(23):234101
Patra PK, Batra RC (2017) Nonequilibrium temperature measurement in a thermal conduction process. Phys Rev E 95:013302
Dhar A (2008) Heat transport in low-dimensional systems. Adv Phys 57:457–537
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The authors acknowledge the support in part provided by the Indian Institute of Technology Kharagpur, through the ISIRD grant DNI.
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Bhattacharyya, S., Patra, P.K. (2022). Effect of Collisions on Properties of Nonequilibrium Steady State of Harmonic Chains with Alternating Masses. In: Maity, D., et al. Recent Advances in Computational and Experimental Mechanics, Vol—I. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-16-6738-1_16
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DOI: https://doi.org/10.1007/978-981-16-6738-1_16
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