%0 Book Section %1 statphys23_0933 %A Hondou, T. %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 demon equation maxwell model molecular motors one-molecule paradox state statphys23 szilard thermodynamics topic-1 %T Equation of state in a small system: Violation of an assumption of Maxwell's demon %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=933 %X An equation of state for an ideal gas with a small number of particles is studied. The equation of state is a basis both for studies of Maxwell's demon paradox since Szilard (1929), including Landauer's principle and for the proper interpretation and modeling of one-molecule experiment in biophysics. It has been assumed, even for the studies in Maxwell's demon, conventional equation of state, $PV=nRT$ holds for systems of small number of particles (for review, see Maxwell's Demon 2 by Leff and Rex, IOP Publishing, 2003). We study the validity of the equation of state in small systems toward the construction of one-molecule thermodynamics. We consider a one-dimensional piston in thermal equilibrium, where the dynamical variables are $N$ thermal particles and a movable piston on one side (Figure 1). In systems of small number of particles, coarse-grained variable, $P$ (pressure) in the equation of state is no longer well-defined. Thus we discuss the validity of the equation of state in the following form, $f X=N k_B T$, in which $f$, $X$ are a mechanical force externally exerted on the movable piston and a distance between the bottom and the movable piston, respectively (Fig. 1). In order to discuss the validity of the equation of state, we first discuss the extreme case of one thermal molecule in the piston, where the total energy of the system is written as $E=p_t^22m + p_p^22M + f X$, where $m (p_t) $ and $M (p_p)$ are the masses (momenta) of the thermal particle and the piston, respectively. To keep the system isothermal, a thermal wall is introduced. The particle is reflected with a positive random velocity according to Maxwell's velocity distribution at the bottom of the piston. One naively expects that the equation of state for one molecule should be obtained by replacing $N$ by $1$ in the reference equation $f X = N k_B T$; namely $f X = k_B T$. The extension of conventional thermodynamics into those of one molecule has been naively performed since Szilard (Z. Phys. 53 p.840, 1929) until now. We found through numerical simulation, however, that the equation of state for one molecule is in fact $f X = 2 k_B T $, where the angular brackets indicate an average value. This result is obviously different from that which is conventionally assumed. We obtained the same analytical result using the Master equation of the distribution function in a phase space, which consists of Liouville terms and collision terms. A stochastic boundary condition is applied for the system at one end, which corresponds to the thermal wall in the numerical simulation. From straightforward calculation, one obtains a stationary solution, $(x, p_t, X, p_p) = f^22 (k_B T)^3 mM \exp\-(p_t^2/2m + p_p^2 /2M + f X k_B T) \ (X-x) ,$ where $þeta$ is a Heaviside step function. From this equation, we again obtain the same equation of state as by the numerical simulation. The result is independent of the masses of both the thermal particle and the movable piston, which is in contrast to the result by Hatano-Sasa (Prog. Theor. Phys. 100 p.695, 1998). Although we introduced a thermal wall and, correspondingly, a stochastic boundary condition, one may obtain the same generalized result for any number of thermal particles, N, without applying such boundary conditions, but instead using conventional Gibbs' statistical mechanics. The result is found to be the same as the conventional equation of state in the thermodynamic limit ($N ınfty$). Impact of our finding on the studies of Maxwell's demon (Szilard's and the followers', including Landauer's) and on one-molecule experiments in biophysics will be discussed in the conference.

@incollection{statphys23_0933, abstract = {An equation of state for an ideal gas with a small number of particles is studied. The equation of state is a basis both for studies of Maxwell's demon paradox [since Szilard (1929), including Landauer's principle] and for the proper interpretation and modeling of one-molecule experiment in biophysics. It has been assumed, even for the studies in Maxwell's demon, conventional equation of state, $PV=nRT$ holds for systems of small number of particles (for review, see {\em Maxwell's Demon 2} by Leff and Rex, IOP Publishing, 2003). We study the validity of the equation of state in small systems toward the construction of one-molecule thermodynamics. We consider a one-dimensional piston in thermal equilibrium, where the dynamical variables are $N$ thermal particles and a movable piston on one side (Figure 1). In systems of small number of particles, coarse-grained variable, $P$ (pressure) in the equation of state is no longer well-defined. Thus we discuss the validity of the equation of state in the following form, $f X=N k_{\rm B} T$, in which $f$, $X$ are a mechanical force externally exerted on the movable piston and a distance between the bottom and the movable piston, respectively (Fig. 1). In order to discuss the validity of the equation of state, we first discuss the extreme case of one thermal molecule in the piston, where the total energy of the system is written as $E=\frac{p_{\rm t}^{2}}{2m} + \frac{p_{\rm p}^{2}}{2M} + f X$, where $m (p_{\rm t}) $ and $M (p_{\rm p})$ are the masses (momenta) of the thermal particle and the piston, respectively. To keep the system isothermal, a thermal wall is introduced. The particle is reflected with a positive random velocity according to Maxwell's velocity distribution at the bottom of the piston. One naively expects that the equation of state for one molecule should be obtained by replacing $N$ by $1$ in the reference equation $f X = N k_{\rm B} T$; namely $f X = k_{\rm B} T$. The extension of conventional thermodynamics into those of one molecule has been naively performed since Szilard (Z. Phys. {\bf 53} p.840, 1929) until now. We found through numerical simulation, however, that the equation of state for one molecule is in fact $f \langle X \rangle = 2 k_{\rm B} T $, where the angular brackets indicate an average value. This result is obviously different from that which is conventionally assumed. We obtained the same analytical result using the Master equation of the distribution function in a phase space, which consists of Liouville terms and collision terms. A stochastic boundary condition is applied for the system at one end, which corresponds to the thermal wall in the numerical simulation. From straightforward calculation, one obtains a stationary solution, $\rho (x, p_{\rm t}, X, p_{\rm p}) = \frac{f^2}{2 \pi (k_{\rm B} T)^3 \sqrt{mM}} \exp\{-(\frac{p_{\rm t}^{2}/2m + p_{\rm p}^2 /2M + f X }{k_{\rm B} T}) \} \theta (X-x) ,$ where $\theta$ is a Heaviside step function. From this equation, we again obtain the same equation of state as by the numerical simulation. The result is independent of the masses of both the thermal particle and the movable piston, which is in contrast to the result by Hatano-Sasa (Prog. Theor. Phys. {\bf 100} p.695, 1998). Although we introduced a thermal wall and, correspondingly, a stochastic boundary condition, one may obtain the same generalized result for any number of thermal particles, N, without applying such boundary conditions, but instead using conventional Gibbs' statistical mechanics. The result is found to be the same as the conventional equation of state in the thermodynamic limit ($N \rightarrow \infty$). Impact of our finding on the studies of Maxwell's demon (Szilard's and the followers', including Landauer's) and on one-molecule experiments in biophysics will be discussed in the conference.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Hondou, T.}, biburl = {https://www.bibsonomy.org/bibtex/29db770b3e327cd9b24156428f3aceb2b/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {d39c60ba04cb9ffaaf9f5fb669e06f1e}, intrahash = {9db770b3e327cd9b24156428f3aceb2b}, keywords = {demon equation maxwell model molecular motors one-molecule paradox state statphys23 szilard thermodynamics topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:34.000+0200}, title = {Equation of state in a small system: Violation of an assumption of Maxwell's demon}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=933}, year = 2007 }