On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications

We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared with traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.

[1]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[2]  A. Frangi,et al.  On the application of the BGK kinetic model to the analysis of gas-structure interactions in MEMS , 2007 .

[3]  Thomas M. M. Homolle,et al.  Low-variance deviational simulation Monte Carlo , 2007 .

[4]  N. Hadjiconstantinou,et al.  Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels 1 , 2002 .

[5]  M. Lundstrom.  Fundamentals of carrier transport , 1990 .

[6]  Lowell L. Baker,et al.  Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation , 2008 .

[7]  A. Manela,et al.  On the motion induced in a gas confined in a small-scale gap due to instantaneous boundary heating , 2007, Journal of Fluid Mechanics.

[8]  Lowell L. Baker,et al.  Variance reduction for Monte Carlo solutions of the Boltzmann equation , 2005 .

[9]  R. Peterson.  Direct simulation of phonon-mediated heat transfer in a Debye crystal , 1994 .

[10]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[11]  G. Bird.  Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .

[12]  A. Manela,et al.  Gas motion induced by unsteady boundary heating in a small-scale slab , 2008 .

[13]  Alejandro L. Garcia,et al.  Statistical error in particle simulations of hydrodynamic phenomena , 2002, cond-mat/0207430.

[14]  Gang Chen,et al.  Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales , 2002 .

[15]  C. Pagani,et al.  Comparison of kinetic theory analyses of linearized heat transfer between parallel plates , 1967 .

[16]  Certain anomalies in the analysis of hyperbolic heat conduction , 2002 .

[17]  Gregg A Radtke,et al.  Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  V. Gurevich.  Transport in phonon systems , 1986 .

[19]  A. Majumdar.  Microscale Heat Conduction in Dielectric Thin Films , 1993 .

[20]  Thomas M. M. Homolle,et al.  A low-variance deviational simulation Monte Carlo for the Boltzmann equation , 2007, J. Comput. Phys..

[21]  E. Richard Cohen,et al.  Neutron Transport Theory , 1959 .

[22]  Nicolas G. Hadjiconstantinou,et al.  Dissipation in Small Scale Gaseous Flows , 2003 .

[23]  A. Majumdar,et al.  Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization , 2001 .

[24]  M. Modest.  Radiative heat transfer , 1993 .

[25]  Adrian F. M. Smith,et al.  Efficient generation of random variates via the ratio-of-uniforms method , 1991 .

[26]  Nicolas G. Hadjiconstantinou,et al.  The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics , 2005 .

[27]  Nicolas G. Hadjiconstantinou,et al.  Analysis of dis-cretization in the direct simulation Monte Carlo , 2000 .

[28]  Carlo Cercignani,et al.  Mathematical Methods in Kinetic Theory , 1970 .

[29]  Lowell L. Baker,et al.  Variance-Reduced Particle Methods for Solving the Boltzmann Equation , 2008 .