General Definitions, Liouville’s Theorem

III. Kinetic Theory of Gases III.A General Definitions • Kinetic theory studies the macroscopic properties of large numbers of particles, start­ ing from their (classical) equations of motion. Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work, heat, and entropy. The phenomenological laws of thermody­ namics tell us how these quantities are constrained as a system approaches its equilibrium. At the microscopic level, we know that these systems are composed of particles (atoms, molecules), whose interactions and dynamics are reasonably well understood in terms of more fundamental theories. If these microscopic descriptions are complete, we should be able to account for the macroscopic behavior, i.e. derive the laws governing the macro­ scopic state functions in equilibrium. Kinetic theory attempts to achieve this objective. In particular, we shall try to answer the following questions: (1) How can we define “equilibrium” for a system of moving particles? (2) Do all systems naturally evolve towards an equilibrium state? (3) What is the time evolution of a system that is not quite in equilibrium? The simplest system to study, the veritable work–horse of thermodynamics, is the dilute (nearly ideal) gas. A typical volume of gas contains of the order of 1023 particles. Kinetic theory attempts to deduce the macroscopic properties of the gas from the time evolution of the individual atomic coordinates. At any time t, the microstate of a system of N particles is described by specifying the positions ~qi (t), and momenta p~i (t), of all particles. The microstate thus corresponds to a point µ(t), in the 6N dimensional phase �N space Γ = i=1 {~qi , ~pi }. The time evolution of this point is governed by the canonical equations  ∂~qi ∂H   =  ∂t ∂~pi  ∂~p ∂H   i =− ∂t ∂~qi , (III.1) where the Hamiltonian H(p, q), describes the total energy in terms of the set of coordinates q ≡ {~q1 , ~q2 , · · · , ~qN }, and momenta p ≡ {p~1 , ~p2 , · · · , ~pN }. The microscopic equations of motion have time reversal symmetry, i.e. if all the momenta are suddenly reversed, p → −p, at t = 0, the particles retrace their previous trajectory, q(t) = q(−t). This follows from the invariance of H under the transformation T (p, q) → (−p, q). 41 As formulated within thermodynamics, the macrostate M , of an ideal gas in equi­ librium is described by a small number of state functions such as E, T , P , and N . The space of macrostates is considerably smaller than the phase space spanned by microstates. Therefore, there must be a very large number of microstates µ corresponding to the same macrostate M . This many to one correspondence suggests the introduction of a statistical ensemble of microstates. Consider N copies of a particular macrostate, each described by a different representative point µn (t), in the phase space Γ. Let dN (p, q, t) equal the number of �N representative points in an infinitesimal volume dΓ = i=1 d3 ~pi d3 q~i around the point (p, q). A phase space density ρ(p, q, t) is then defined from ρ(p, q, t)dΓ = lim N →∞ dN (p, q, t) . N (III.2) This quantity can be compared with the objective probability introduced in the previous � section. Clearly dΓρ = 1, and ρ is a properly normalized probability density function in phase space. To compute macroscopic values for various functions O(p, q), we shall use the ensemble averages �O� = � dΓρ(p, q, t)O(p, q). (III.3) When the exact microstate µ is specified, the system is said to be in a pure state. On the other hand, when our knowledge of the system is probabilistic, in the sense of its being taken from an ensemble with density ρ(Γ), it is said to belong to a mixed state. It is difficult to describe equilibrium in the context of a pure state, since µ(t) is constantly changing in time according to eqs.(III.1). Equilibrium is more conveniently described for mixed states by examining the time evolution of the phase space density ρ(t), which is governed by the Liouville’s equation introduced in the next section. III.B Liouville’s Theorem • Liouville’s Theorem states that the phase space density ρ(Γ, t), behaves like an incompressible fluid. Proof: Follow the evolution of dN pure states in an infinitesimal volume dΓ = �N 3 3 ~i d ~qi around the point (p, q). According to eqs.(III.1), after an interval δt these i=1 d p states have moved to the vicinity of another point (p′ , q′ ), where qα′ = qα + q̇α δt + O(δt2 ) , 42 p′α = pα + ṗα δt + O(δt2 ). (III.4) In the above expression, the qα and pα refer to any of the 6N coordinates and momenta, and q̇α and ṗα are the corresponding velocities. The original volume element dΓ, is in the shape of a hyper-cube of sides dpα and dqα . In the time interval δt it gets distorted, and the projected sides of the new volume element are given by  ∂q̇α  ′  dqα δt + O(δt2 )  dqα =dqα + ∂qα  ′ ∂ṗα   dpα δt + O(δt2 ) dpα =dpα + ∂pα To order of δt2 , the new volume element is dΓ′ = follows that for each pair of conjugate coordinates dqα′ · dp′α � = dqα · dpα 1 + � �N i=1 ∂ṗα ∂q̇α + ∂pα ∂qα � . (III.5) d3 p~i ′ d3 q~i ′ . From eqs.(III.5) it 2 � δt + O(δt ) . (III.6) But since the time evolution of coordinates and momenta are governed by the canonical eqs.(III.1), we have ∂ ∂H ∂2H ∂q̇α = = , ∂qα ∂pα ∂pα ∂qα ∂qα and ∂ṗα ∂ = ∂pα ∂pα � � ∂H ∂ 2H − =− . ∂qα ∂qα ∂pα (III.7) Thus the projected area in eq.(III.6) is unchanged for any pair of coordinates, and hence the volume element is unaffected, dΓ′ = dΓ. All the pure states dN , originally in the vicinity of (p, q) are transported to the neighborhood of (p′ , q′ ), but occupy exactly the same volume. The ratio dN /dΓ is left unchanged, and ρ behaves like the density of an incompressible fluid. The incompressibility condition ρ(p′ , q′ , t + δt) = ρ(p, q, t), can be written in differ­ ential form as 3N dρ ∂ρ � = + dt ∂t α=1 � ∂ρ dqα ∂ρ dpα · + · ∂qα dt ∂pα dt � = 0. (III.8) Note the distinction between ∂ρ/∂t and dρ/dt: The former partial derivative refers to the changes in ρ at a particular location in phase space, while the latter total derivative follows the evolution of a volume of fluid as it moves in phase space. Substituting from eq.(III.1) into eq.(III.8) leads to � 3N � ∂ρ � ∂ρ ∂H ∂ρ ∂H = · − · = −{ρ, H}, ∂qα ∂pα ∂t ∂pα ∂qα α=1 43 (III.9) where we have introduced the Poisson bracket of two functions in phase space as {A, B} ≡ 3N � � ∂A α=1 ∂B ∂A ∂B · − · ∂pα ∂qα ∂qα ∂pα � = −{B, A}. (III.10) (1) Under the action of time reversal, (p, q, t) → (−p, q, −t), the Poisson bracket {ρ, H} changes sign, and eq.(III.9) implies that the density reverses its evolution, i.e. ρ(p, q, t) = ρ(−p, q, −t). (2) The time evolution of the ensemble average in eq.(III.3) is given by (using eq.(III.9)) d �O� = dt � � � 3N � � ∂ρ(p, q, t) ∂ρ ∂H ∂ρ ∂H dΓ O(p, q) = dΓO(p, q) · − · . ∂qα ∂pα ∂t ∂pα ∂qα α=1 (III.11) The partial derivatives of ρ in the above equation can be removed by using the method � ′ � of integration by parts, i.e. f ρ = − ρf ′ since ρ vanishes on the boundaries of the integrations, leading to �� � � 2 �� 3N � � ∂ 2H d �O� ∂O ∂H ∂O ∂H ∂ H =− dΓρ · − · +O − ∂qα ∂pα ∂pα ∂qα ∂qα ∂pα dt ∂pα ∂qα α=1 (III.12) � = − dΓρ{H, O} = �{O, H}� . Note that the total time derivative cannot be taken inside the integral sign, i.e. � d �O� dρ(p, q, t) = � dΓ O(p, q). dt dt (III.13) This common mistake yields d �O� /dt = 0! (3) If the members of the ensemble correspond to an equilibrium macroscopic state, the ensemble averages must be independent of time. This can be achieved by a stationary density, ∂ρeq /∂t = 0, i.e. by requiring {ρeq , H} = 0. (III.14) A possible solution to the above equation is for ρeq to be a function of H, i.e. ρeq (p, q) = � � ρ H(p, q) . It is then easy to verify that {ρ(H), H} = ρ′ (H){H, H} = 0. This solution implies that the value of ρ is constant on surfaces of constant energy H, in phase space. This is indeed the basic assumption of statistical mechanics. For example, in the microcanonical ensemble, the total energy E of an isolated system is specified. All members of the ensemble 44 must then be located on the surface H(p, q) = E in phase space. Eq.(III.9) implies that a uniform density of points on this surface is stationary in time. The assumption of statistical mechanics is that the macrostate is indeed represented by such a uniform density of microstates. This is equivalent to replacing the objective measure of probability in eq.(III.2) with a subjective one. There may be additional conserved quantities associated with the Hamiltonian which satisfy {Ln , H} = 0. In the presence of such quantities, a stationary density exists for any � � function of the form ρeq (p, q) = ρ H(p, q), L1 (p, q), L2 (p, q), · · · . Clearly, the value of Ln is not changed during the evolution of the system, since dLn (p, q) Ln (p(t + dt), q(t + dt)) − Ln (p(t), q(t)) ≡ dt dt � � 3N � ∂Ln ∂pα ∂Ln ∂qα = · + · ∂qα ∂t ∂pα ∂t α=1 � 3N � � ∂Ln ∂H ∂Ln ∂H =− · − · = {Ln , H} = 0. ∂q ∂q ∂p ∂p α α α α α=1 (III.15) Hence, the functional dependence of ρeq on these quantities merely indicates that all ac­ cessible states, i.e. those that can be connected without violating any conservation law, are equally likely. (4) The above postulate for ρeq answers the first question posed at the beginning of this chapter. However, in order to answer the second question, and to justify the basic assumption of statistical mechanics, we need to show that non-stationary densities converge onto the stationary solution ρeq . This contradicts the time reversal symmetry noted in (1) above: For any solution ρ(t) converging to ρeq , there is a time reversed solution that diverges from it. The best that can be hoped for is to show that the solutions ρ(t) are in the neighborhood of ρeq the majority of the time, so that time averages are dominated by the stationary solution. This brings us to the problem of ergodicity, which is whether it is justified to replace time averages with ensemble averages. In measuring the properties of any system, we deal with only one representative of the equilibrium ensemble. However, most macroscopic properties do not have instantaneous values and require some form of averaging. For example, the pressure P exerted by a gas results from the impact of particles on the walls of the container. The number and momenta of these particles varies at different times and different locations. The measured pressure reflects an average over many characteristic microscopic times. If over this time scale the representative point of 45 the system moves around and uniformly samples the accessible points in phase space, we may replace the time average with the ensemble average. For a few systems it is possible to prove an ergodic theorem, which states that the representative point comes arbitrarily close to all accessible points in phase space after a sufficiently long time. However, the proof usually works for time intervals that grow exponentially with the number of particles N , and thus exceed by far any reasonable time scale over which the pressure of a gas is typically measured. As such the proofs of the ergodic theorem have so far little to do with the reality of macroscopic equilibrium. 46