%0 Book Section %1 statphys23_0618 %A Giuggioli, L. %A Parris, P.E. %A Kenkre, V.M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K approximation boltzmann equation mobility relaxation statphys23 time topic-1 %T Variational Determination of a Relaxation Time to Equilibrium in the Boltzmann equation %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=618 %X The Boltzmann equation, the work-horse of transport calculations in gas dynamics, solid state physics, and related branches of investigation, is essential to the analysis of transport coefficients such as viscosity, electrical mobility, thermal conductivity, Peltier coefficients and Lorenz numbers. The equation comes in two forms, one linear in the probability distribution (or density) $f_k(t)$, the other bilinear. The former is the so-called linearized Boltzmann equation, also known, in other contexts, as the gain-loss equation, and it is the subject of our study here: equation df_kłeft( t\right) dt=\sum_k^łeft Q_kk^\prime f_k^łeft( t\right) -Q_k^kf_kłeft( t\right) % \right. Boltzmann equation Here $k$ represents generally a quantum mechanical state and $Q_k^\prime k$ is the transition rate from state $k$ to state $k^$ and is independent of the $f$'s. Although linear, Eq. (1) is not easily solved because of the summation in $k$-space which in the continuum limit would make (1) an integral equation. There is, however, an approximation, traditionally used in many, if not all, practical applications, that allows one to avoid solving Eq. (1) numerically: the so-called relaxation time approximation (RTA). In its most common form, the RTA consists of replacing the actual evolution in Eq. (1) by equation df_kłeft( t\right) dt+f_kłeft( t\right) -f_k^th% _k=0, relax equation where $\tau_k$ is called the relaxation time and $f_k^th$ is the thermal form to which the distribution tends at long times in the absence of driving forces. Although there are variants, the simplest and most common prescription for the relaxation time that one finds in the literature is equation 1_k=\sum_k^Q_k^k. tau equation Despite its common use, the RTA suffers from the main drawbacks that it is independent of initial conditions as well as for not conserving probability at all times. Using techniques from the calculus of variations we derive a variational principle for the Boltzmann equation and use it to obtain the approximation equation f_kłeft( t\right) =f_k^th+a_ke^-t/\tau ansatzvar equation with $$ 1/\tau=\frac\sum_kłeft(\sum_k^\primekQ_k^\prime kf_k(0)-\sum_k^\primekQ_kk^\primef_k^\prime (0)\right)^2/f_k^th\sum_kłeft(f_k(0)-f_k^th\right) ^2/f_k^th. varrate $$ It is obvious that Eq. (4) conserves probability at all times and allows us to extend the standard RTA using relaxation times that depend on the initial distribution. Tests of the approach on a calculation of the mobility of a particle moving in a 1D tight binding band indicate that our analysis provides a better approximation than the standard RTA. Reference: L. Giuggioli, P.E. Parris and V.M. Kenkre, J. Phys. Chem. B 110, 18921 (2006).

@incollection{statphys23_0618, abstract = {The Boltzmann equation, the work-horse of transport calculations in gas dynamics, solid state physics, and related branches of investigation, is essential to the analysis of transport coefficients such as viscosity, electrical mobility, thermal conductivity, Peltier coefficients and Lorenz numbers. The equation comes in two forms, one linear in the probability distribution (or density) $f_{k}(t)$, the other bilinear. The former is the so-called linearized Boltzmann equation, also known, in other contexts, as the gain-loss equation, and it is the subject of our study here: \begin{equation} \frac{df_{k}\left( t\right) }{dt}=\sum_{k^{\prime }}\left[ Q_{kk^{\prime }}f_{k^{\prime }}\left( t\right) -Q_{k^{\prime }k}f_{k}\left( t\right) % \right]. \label{Boltzmann} \end{equation} Here $k$ represents generally a quantum mechanical state and $Q_{k^{\prime }k}$ is the transition rate from state $k$ to state $k^{\prime }$ and is independent of the $f$'s. Although linear, Eq. (1) is not easily solved because of the summation in $k$-space which in the continuum limit would make (1) an integral equation. There is, however, an approximation, traditionally used in many, if not all, practical applications, that allows one to avoid solving Eq. (1) numerically: the so-called relaxation time approximation (RTA). In its most common form, the RTA consists of replacing the actual evolution in Eq. (1) by \begin{equation} \frac{df_{k}\left( t\right) }{dt}+\frac{f_{k}\left( t\right) -f_{k}^{th}}{% \tau _{k}}=0, \label{relax} \end{equation} where $\tau_{k}$ is called the relaxation time and $f_{k}^{th}$ is the thermal form to which the distribution tends at long times in the absence of driving forces. Although there are variants, the simplest and most common prescription for the relaxation time that one finds in the literature is \begin{equation} \frac{1}{\tau _{k}}=\sum_{k^{\prime }}Q_{k^{\prime }k}. \label{tau} \end{equation} Despite its common use, the RTA suffers from the main drawbacks that it is independent of initial conditions as well as for not conserving probability at all times. Using techniques from the calculus of variations we derive a variational principle for the Boltzmann equation and use it to obtain the approximation \begin{equation} f_{k}\left( t\right) =f_{k}^{th}+a_{k}e^{-t/\tau} \label{ansatzvar} \end{equation} with $$ 1/\tau=\sqrt{\frac{\sum_{k}\left(\sum_{k^{\prime}\neq k}Q_{k^{\prime }k}f_{k}(0)-\sum_{k^{\prime}\neq k}Q_{kk^{\prime}}f_{k^{\prime }}(0)\right)^{2}/f_{k}^{th}}{\sum_{k}\left(f_{k}(0)-f_{k}^{th}\right) ^{2}/f_{k}^{th}}}. \label{varrate} $$ It is obvious that Eq. (4) conserves probability at all times and allows us to extend the standard RTA using relaxation times that depend on the initial distribution. Tests of the approach on a calculation of the mobility of a particle moving in a 1D tight binding band indicate that our analysis provides a better approximation than the standard RTA. Reference: L. Giuggioli, P.E. Parris and V.M. Kenkre, J. Phys. Chem. B 110, 18921 (2006).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Giuggioli, L. and Parris, P.E. and Kenkre, V.M.}, biburl = {https://www.bibsonomy.org/bibtex/269c0aeef64b1004afbe7674f13f50ccb/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {660b4848ccf3d8f6751da56fd4e837db}, intrahash = {69c0aeef64b1004afbe7674f13f50ccb}, keywords = {approximation boltzmann equation mobility relaxation statphys23 time topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:25.000+0200}, title = {Variational Determination of a Relaxation Time to Equilibrium in the Boltzmann equation}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=618}, year = 2007 }