\documentstyle[prl,aps]{revtex} \begin{document} \draft % \preprint{ } \title{A Dynamical Approach to Temperature} \author{Hans Henrik Rugh } \address {Department of Mathematics, University of Warwick, Coventry, CV4 7AL, England} \date{\today} \twocolumn[ % omit in submit \maketitle \widetext % omit in submit \begin{abstract} We present a new dynamical approach for measuring the temperature of a Hamiltonian dynamical system in the micro canonical ensemble of thermodynamics. We show that under the hypothesis of ergodicity the temperature can be computed as a time-average of the functional, $\nabla \cdot (\nabla H / \|\nabla H\|^2)$, on the energy-surface. Our method not only yields an efficient computational approach for determining the temperature it also provides an intrinsic link between dynamical systems theory and the statistical mechanics of Hamiltonian systems. Phys. Rev. Lett (vol 78, no. 4), 772-774 (1997). \end{abstract} \pacs{05.20.Gg, 05.20.-y, 05.45.+b, 05.70.-a, 02.40.Vh, 02.40.-k} ] \narrowtext % omit in submit Two crucial assumptions allow one to pass from the dynamical to the thermodynamical description of a classical Hamiltonian system. First, the system should exhibit some sort of ergodicity so that time-averages can be replaced with equivalent space-averages over the micro-canonical ensemble, i.e. over the corresponding energy surface (possibly with other constraints) and with respect to the invariant Liouville measure. Second, it should make sense to consider a large number of (weakly) coupled sub-systems and look at the statistical behavior of an individual fixed sub-system. As the size of the ambient system tends to infinity one obtains the thermodynamic limit in which the micro-canonical ensemble involving the sub-system is assumed to behave like the canonical or the Gibbs ensemble where quantities like temperature and pressure become operationally defined. In a few special cases it has been possible to prove the equivalence of the two ensembles but it is believed that under quite general assumptions the two ensembles indeed give equivalent descriptions when systems are large. We refer to Ruelle \cite{Ruelle} for a more complete treatment on the equivalence of ensembles, to Lebowitz et al. \cite{LPV} for an illustrative example of some differences between the ensembles and to Landau and Lifshitz \cite {LL} for a general introduction to the subject. For practical calculations using the micro canonical ensemble see e.g. Evans and Morriss \cite{EM}. In the thermodynamic description the temperature plays a prominent role and the problem of measuring it using the dynamical evolution of the system dates back to the early days of thermodynamics. Already Maxwell realized that when the Hamiltonian has the special form $H(p,q,\ldots) = p^2/2 + K(q,\cdots)$, thus singling out the square of a momenta, the canonical ensemble average of $p^2$ happens to be the temperature. Thus assuming equivalence of ensembles and ergodicity it suffices to measure the time-average of $p^2$ during the dynamical evolution in order to get the temperature. Here we shall pursue a different approach in which we focus on the global geometric structure of the energy surface We shall assume ergodicity only and show that the temperature as defined in the micro-canonical ensemble can be obtained as a purely dynamical time-average of a function which is related to the curvatures of the energy surface. In this presentation we shall for clarity and without loss of generality restrict ourselves to a Euclidean phase space\footnote{ The Euclidean metric is not necessary but provides a convenient way of expressing the dynamical average in terms of gradients of functions and divergences of vector fields.}, $\Omega = R^{2d}$\ $(d \geq 1)$, and a Hamiltonian function, $H : \Omega \rightarrow R$, which preserves the Liouville measure (here Lebesgue measure $m$) and such that for all energies considered the set\ $H(\xi) \leq E$\ is bounded and has a smooth and connected surface, $A(E) = \{H(\xi)=E\}$. Furthermore, we shall consider the case when the energy itself, $H=E$, is the only preserved first integral. We refer to \cite{Rugh} for a generalization of this approach. For the micro-canonical ensemble of classical thermodynamics one defines the exponential of the entropy to be the (canonically invariant) weighted area of an energy surface~: \begin{equation} e^{S(E)} \equiv \int m(d\xi) \; \delta \left( H(\xi)\!-\!E \right) = \int_{A(E)} \frac{ m(dA_\xi)}{\|\nabla H(\xi)\|} , \label{eq:entropy} \end{equation} and the temperature is obtained through differentiation \begin{equation} \frac{1}{T(E)} = \frac{dS}{dE} . \label{eq:temp} \end{equation} In practise a calculation of the entropy function is impossible when the number of degrees of freedom in the system is large. However, we have the following~:\\ \noindent \underline{Theorem :} {\em Suppose that $(\Omega,H)$ is ergodic with respect to the Liouville measure $m_{|A(E)}$, restricted to a non-singular energy surface, $A(E)$. Then for $m_{|A(E)}$-almost every $x(0)\in A(E)$, one has \begin{equation} \frac{1}{T(E)} = \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t d\tau\; \Phi(x(\tau)) , \label{eq:dyntemp} \end{equation} where $x(\tau)$ is the trajectory of $x(0)$ under the Hamiltonian flow and the observable $\Phi$ equals $\nabla \cdot (\nabla H / \|\nabla H\|^2)$, which can also be written as~: \begin{equation} \Phi = \frac{\Delta H}{\| \nabla H\|^2} - 2 \frac{D^2 H (\nabla H, \nabla H)}{\| \nabla H\|^4} . \label{eq:Phi} \end{equation} } Remarks :\\[-3mm] \begin{enumerate} \item The measure and the energy are canonical invariants. Hence, from equations (\ref{eq:entropy}) and (\ref{eq:temp}), so are the temperature and the time-average, equation (\ref{eq:dyntemp}), of $\Phi$ but {\em not} $\Phi$ itself. The theorem shows also that the temperature unlike the entropy is an intrinsic dynamical feature of the system (it can be computed as a dynamical average). \item The theorem can quite easily, at the cost of making the notation more heavy, be generalized to arbitrary symplectic manifolds without a Riemannian structure (cf. \cite{Rugh}). \item It is of interest to see how the conjugated thermodynamic variables of other preserved first-integrals (e.g. total angular momentum) can be incorporated in the above dynamical formalism. It is for example well-known that the pressure can be obtained as a time-average of the momentum transfer per surface area element in the configuration space. It is also of interest to derive general \mbox{(thermo-)}dynamical relations for such quantities in the micro canonical ensemble (cf. \cite{Rugh}). \item There is a formal resemblance of $\Phi$ to the Gauss curvature $1/R = \nabla \cdot (\nabla H / \|\nabla H\|)$, where for the latter quantity the norm of the gradient is not squared. In particular, when the number of degrees of freedom gets large (and the last term in the formula (\ref{eq:Phi}) becomes relatively small) the relationship becomes closer. We do not know of any deeper reasons for this connection. \end{enumerate} The idea behind the above Theorem is to cast the energy dependence of the geometry, i.e. the dependence on $A(E)$ in equations (\ref{eq:entropy}) and (\ref{eq:temp}), into an algebraic dependence for a fixed geometry. This is achieved through the introduction of an auxiliary vector field (to a large extend arbitrary) and it will enable us to take the derivative with respect to the energy inside the integral. We consider in the following two vector fields on $\Omega$~: \begin{equation} v = J \nabla H \ \ \ \mbox{and} \ \ \ \eta = \frac{\nabla H} {\langle {\nabla H,\nabla H} \rangle} . \end{equation} Here $J$ is a symplectic matrix, e.g. $\left(\begin{array}{rr}\bf{0} & \bf{1}\\-\bf{1} & \bf{0} \end{array} \right)$, and $\langle {\cdot,\cdot} \rangle$ is the usual Euclidean product in $\Omega$. For a vector field $X$ on $\Omega$ we let $g^t_X$ denote the associated flow. Thus $g^t_v$ is the Hamiltonian flow and $g^s_\eta$ is the normalized energy gradient flow. Since $\frac{d}{ds} H ( g^s_\eta(x)) = \langle{\nabla H(x),\eta(x)}\rangle = 1$ we have \begin{equation} H ( g^s_\eta(x)) = s + H(x) . \end{equation} Assuming that no singular points are encountered along the gradient flow, i.e. that $\nabla H \neq 0$ everywhere on the energy surface (which is a generic condition), the map \begin{equation} g^s_\eta : A(E) \rightarrow A(E+s) \label{eq:diffeo} \end{equation} is a diffeomorphism between energy surfaces. Both the Liouville measure $m$ and the energy are preserved under the Hamiltonian flow, i.e. $m(g^t_v dx)=m(dx)$ and $H(g^t_v(x))=H(x)$. It follows that by considering an infinitesimal flow-box (under the energy gradient flow) $g^{[0,\epsilon]}_\eta dA$ and by measuring the volume of such a flow-box we may define a measure $\mu$ on the energy surface $A(E)$ which is also invariant under $g^t_v$, \begin{equation} \mu(dA) = \lim_{\epsilon\rightarrow 0^+} \frac{1}{\epsilon} m \{ g^{[0,\epsilon]}_\eta dA \} . \end{equation} For this measure, we have $e^{S(E)} = \int_{A(E)} \mu(dA)$. The flow $g^s_\eta$ commutes with itself and hence it maps the flow-box $g^{[0,\epsilon]}_\eta dA(E)$ into the flow-box $g^{[0,\epsilon]}_\eta dA(E+s)$ (by equation \ref{eq:diffeo}). But then the ratios between the weighted areas of infinitesimal surface elements just becomes a ratio between volume elements which is nothing but the Jacobian of the mapping~: \[ \frac{\mu(dA(E+s))}{\mu(dA(E))} = \det (\frac{ \partial g^s_{\eta}}{\partial x} ) = 1 + s\; \nabla \cdot \eta + O(s^2) \] (to order $s$). Defining $\Phi = \nabla \cdot \eta$ (i.e. our equation \ref{eq:Phi}) we get the relation~: \begin{equation} \int_{A(E+s)} \mu(dA) = \int_{A(E)} \mu(dA) (1+ s \Phi + ...) , \end{equation} and finally, using the definition of temperature (\ref{eq:temp}) we obtain~: \begin{equation} \frac{1}{T(E)} = \frac{\int_{A(E)} \mu(dA) \Phi} {\int_{A(E)} \mu(dA)} . \end{equation} The interesting observation is now that if the Hamiltonian flow is ergodic with respect to the Liouville measure restricted to the energy surface, $\mu$, then by Birkhoff's theorem the right hand side equals the time-average (and thus equation \ref{eq:dyntemp}) of $\Phi$ for $\mu$-almost every (equivalent to $m_{|A(E)}$-a.e.) trajectory under the Hamiltonian flow, thus proving our theorem.\\ An illustrative, though trivial\footnote {Strictly speaking this example does not meet the conditions of our Theorem as it is not ergodic.}, example is given if we take $N$ harmonic oscillators with the Hamiltonian $H = \sum_{i=1}^N \frac{1}{2}(q_i^2+p_i^2)$. Its gradient $\nabla H = (\vec{q},\vec{p})$ is non-vanishing at all non-zero energies and a straight forward calculation yields $\Phi = (N-1)/{E} = {1}/{T(E)}$. In this particular case we can directly read off the micro-canonical result, $E = (N-1)T$. The fixation of the energy `freezes' one degree of freedom as is seen by comparing with the canonical result, $\langle E \rangle = NT$. \\ In conclusion we have described a dynamical way of measuring the temperature in the micro canonical ensemble of thermodynamics. It is rigorously shown to work for a Hamiltonian system at energies where the energy surface is regular and the flow is ergodic. The approach is purely dynamical and involves calculating the time-average of the functional, $\nabla \cdot (\nabla H / \|\nabla H\|^2)$, on the energy surface. As the derivation shows the dynamical temperature is really intrinsic to the dynamical system in contrast to the entropy itself. We note that in a classical theory the entropy is in any case only defined up to a constant. We believe that the results and the underlying ideas presented here may be of practical importance in numerical simulations and helpful to physicists who wish to understand the foundations of classical thermodynamics. I am grateful to Predrag Cvitanovi\'c, Robert MacKay, David Ruelle as well as to many referees for \mbox{(thermo-)}dynamic discussions. \begin{thebibliography}{X} \bibitem{EM} D.E. Evans and G.P. Morriss, {\em Statistical Mechanics of non-Equilibrium Liquids}, Acad. Press (1990). \bibitem{LL} L.D. Landau and E.M. Lifshitz, {\em Statistical Physics, Part 1}, Oxford, Pergamon (1980). \bibitem{LPV} J.L. Lebowitz, J.K Percus and L. Verlet, {\em Ensemble Dependence of Fluctuations with Application to Machine Computations}, Phys. Rev. {\bf 153}, 250 (1967). \bibitem{Ruelle} D. Ruelle, {\em Statistical mechanics: Rigorous Results}, Addison-Wesley (1989). \bibitem{Rugh} H.H. Rugh, {\em A Dynamical Approach to Thermodynamics in the $\mu$-Canonical Ensemble} (in preparation). \end{thebibliography} \end{document}