On the Gibbs-Liouville Theorem in Classical Mechanics

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On the Gibbs-Liouville theorem in classical mechanics

Andreas Henriksson

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Abstract In this article, it is argued that the Gibbs- 7 The Hamiltonian and Lagrangian ...... 7 Liouville theorem is a mathematical representation of 8 Principle of stationary action ...... 8 the statement that closed classical systems evolve de- 9 Invariance of the Poisson algebra ...... 9 terministically. From the perspective of an observer of 10 Unitary time-evolution operator ...... 10 the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that 1 Introduction the physical distinctions between the possible states of the system, or, in other words, the information pos- A key concept which is at the heart of the distinction sessed by the observer about the system, is never lost. between the temporal evolution of classical and quan- Furthermore, it is shown that the Hamilton equations tum systems is determinism. Classical systems are said and the Hamilton principle on phase space follow di- to be deterministic while quantum systems are non- rectly from the differential representation of the Gibbs- deterministic. Considering this key distinction, it is the Liouville theorem, i.e. that the divergence of the Hamil- purpose of this article to initiate a study on the role tonian phase flow velocity vanish. Finally, it is argued of determinism in classical and quantum mechanics by that the statements of invariance of the Poisson algebra rephrasing the conventional exposition of classical me- and unitary evolution are equivalent representations of chanics in such a manner that the notion of determin- the Gibbs-Liouville theorem. istic evolution take the central role. Keywords Determinism · Information · Gibbs- The conventional exposition of classical mechanics is Liouville theorem · Hamilton equations · Hamilton largely based on the historical development of the sub- principle · Poisson algebra · Unitarity ject [1][2][3][4]. Newton introduced the concept of force in order to describe the motion of objects, as mathe- PACS 45.20.Jj · 45.05.+x · 45.20.-d · 03.50.-z matically expressed by his second law of motion, which is a second-order differential equation in time. Later, Lagrange [5][6] constructed a mathematically equiva- Contents lent formulation where the Lagrangian function, which is defined on configuration space, is introduced and 1 Introduction ...... 1 which play the central role. The Lagrange, or Euler- 2 Phase space ...... 2 3 Determinism and information ...... 2 Lagrange, equations of motion are also second-order 4 Incompressible fluid flow ...... 4 differential equations in time. Lagrange’s formulation 5 The Gibbs-Liouville theorem ...... 5 was later transformed from configuration space onto 6 Hamilton’s equations ...... 7 phase space by Hamilton [7], where the central role Andreas Henriksson is played by the Hamiltonian function. The Hamilton St.Olav school, Jens Zetlitzgt. 33, 4008, Stavanger; Norway equations of motion are first-order differential equa- E-mail: [email protected] tions in time. The deterministic character must then ORCID: 0000-0001-9014-4320 be proved by showing that the equations of motion, 2 Andreas Henriksson with known unique initial conditions, have unique so- lutions such that the states cannot converge into each other. Thus, determinism enter into the discussion at a fairly late stage, after the introduction and definition of concepts such as force, the Lagrangian and Hamilto- nian functions, and after the equations of motion have been obtained. In this article, the attempt is to start with the notion of determinism and from it deduce the necessary form of the equations of motion. The purpose behind the desire to shift the exposition of classical mechanics as proposed in this article is to, hopefully, better understand the non-deterministic evolution for quantum systems on phase space, as de- Fig. 1 Phase space for a continuous system. scribed in the seminal paper by Moyal [8].

mean that the entire evolutionary history of the system 2 Phase space is known.

The theatrical stage on which physical phenomena are played out are characterized by so-called degrees of freedom. They are parameters whose values are aimed at defining the state of existence for a system as it unfold in time. There exist different mathematical representations of this theatrical stage. In this article, the space of phases, or phase space, is the stage used to study the flow of classical systems. At any given time t, any given particle j within an N−particle system with three spatial dimensions has the spatial location qj ≡ (qj1, qj2, qj3)j and the momenta pj ≡ (pj1, pj2, pj3)j. These are the degrees of freedom for the j0th particle. The total number of degrees of freedom for the system is thus 6N. For nota- tional simplicity, the 3N spatial and the 3N momenta Fig. 2 The extended phase space. degrees of freedom are denoted by q and p, respectively. The space of all possible values for the pair (q, p) define the phase space. Each point in phase space correspond to a specific state for the system, i.e. a specific value for the spatial location and momenta of each particle 3 Determinism and information within the system. For continuous systems, all states in which the sys- In classical mechanics, it is a fundamental assumption tem can exist are continuously connected to each other that the evolution of a system is deterministic in both in the sense that any two arbitrary states can be trans- directions of time, i.e. both into the future and into the formed into each other by considering successive in- past. Deterministic evolution of a system mean that it is finitesimal variations in q and p. Thus, in the contin- possible, with absolute certainty, to say that any given uous case, the states of a system lie on smooth sur- state of the system evolved from a definite single state faces in phase space, see figure 1, where the state at in the past and will evolve into a definite single state time t0, given by the values (q0, p0), is connected to the in the future. There cannot be any ambiguity in the state at another time t, given by the values (q(t), p(t)), evolutionary history of a system. Thus, deterministic along a smooth trajectory, the so-called system trajec- evolution imply that nowhere on phase space can states tory. Phase space can be extended to explicitly include converge or diverge, see figure 3. time as a coordinate which is orthogonal to phase space, Systems that appear to evolve non-deterministically see figure 2. As the system evolve in time, it traces out give rise to the appearance of irreversible processes in a trajectory in the extended phase space. If the tra- nature. The reason for this is that if a system start jectory on phase space is known for a given system, it out in a given state it is not necessarily the case that On the Gibbs-Liouville theorem in classical mechanics 3

perfect track of the motion of all particles. In the case of the sliding block, for a human observer, the distinction between individual motions of atoms in the block and table are too small to measure and therefore it appear as though two distinct initial states, characterized by distinct initial speeds, which are easy to measure, converge to the same ﬁnal state, i.e. that the block is at rest. In conclusion, the assumption of deterministic evolution can equivalently be stated as follows.

The distinction between physical states of a closed system is conserved in time.

Fig. 3 Non-deterministic evolution imply that system tra- Due to the conservation of distinction between phys- jectories would cross each other on phase space, here at point ical states, any set of states which lie in the interior (q , p ). 0 0 of some volume element on phase space will remain interior of this volume element as the system evolve in the system end up at the same initial state by revers- time. ing the motion of the system in time. An example of a If a system is followed, as it evolve in time, in per- seemingly irreversible process is the sliding of a block fect detail by an observer, it mean that the observer has of cheese along a table. Due to friction the block will perfect and complete knowledge about all the degrees always come to rest, apparently independent on the ini- of freedom of the system, i.e. the observer know, with tial condition of the block. Thus it appear as though the infinite precision, the exact position and momenta of all multitude of possible initial states for the block, given particles within the system. In such an ideal scenario, by the possibility of sending off the block with differ- the observer has no problem to see the distinction be- ent initial speeds, all converge to the same final state tween states of the system. The amount of knowledge, where the block is at rest. Knowing the final state of or information, about the system possessed by the ob- the system does not help in predicting the initial state server, at any instant of time, is complete. Since the of the system. Therefore, the experiment with sending ideal observer never loose track of the system, the dis- off the block of cheese seem to represent an evolution tinction between states is never lost. In other words, the which is non-deterministic into the past. knowledge, or information, that the observer has about The origin for the apparent violation of reversibil- the system is not lost as the system evolve in time. ity in physical processes is not due to a fundamental character in physical laws, but rather it is due to the If, however, as is the case in practical reality, the ignorance of the observer. The observer has not taken observer has a limited ability to track the motion of in- into account all the details of the system. Degrees of dividual particles, the observer do not possess complete freedom for the system has been ignored. In the case of information about the system. Even worse, the observer the sliding block of cheese, it is the individual motion may, as is usually the case for complicated systems with of atoms in the block and table which has been ignored. many degrees of freedom, find it more and more difficult Assuming that all degrees of freedom for the block and to track the system as time unfold. In such a scenario, table are followed in perfect detail as the block slide on the amount of information about the system, possessed the table it is clear that each unique initial state will by the observer, decrease with time. In other words, give rise to a unique final state where the distinction from the perspective of the ignorant observer, informa- between the final states are given by the distinct final tion about the system is lost. However, it is important position and velocity of each atom in the block and to emphasize that this apparent loss of information is table. entirely due to the ignorance of the observer. If all the A direct consequence of the assumption of deter- degrees of freedom were tracked with infinite precision, ministic evolution is that distinctions between physical information would never be lost. In the case of the slid- states never disappear. If there is an initial distinction ing block of cheese, the observer has lost information between states, this distinction will survive throughout because the system was known to exist in one of two the entire motion of the system. That distinctions be- distinct initial states, obtained by measuring the initial tween states seem to disappear as time unfold is merely speed of the block, whereas it is not possible to distin- a consequence of the difficulty for an observer to keep guish between the two final states. 4 Andreas Henriksson

In conclusion, the loss of distinction between states This is rewritten as imply that information has been lost. Thus, the con- ∆ (ρ(x)v(x)) ≡ ρ(x )v(x ) − ρ(x )v(x ) = 0 (4) servation of distinction between states can equivalently out out in in be stated as an assumption of information conservation: In differential form it read d d (ρ(x)v(x)) ≡ (ρ(x)v(x)) · ∆x = 0 (5) The information contained within a closed system is dx conserved in time. which gives that In other words, the assumption that classical systems d (ρ(x)v(x)) = 0 (6) evolve deterministically, i.e. that the state of the sys- dx tem is perfectly predictable by an observer both into For the number of molecules to be conserved over an the future and back to the past, is equivalent to the extended period of time, i.e. over many successive time statement that an observer of the system possess com- intervals ∆t, the required condition become plete information about the system, and assuming that ∂ρ(x, t) ∂ the system is closed, this amount of information is never + (ρ(x, t)v(x)) = 0 (7) lost. ∂t ∂x This is the continuity equation for the flow of molecules in one dimension. The product rule on the second term 4 Incompressible fluid flow give ∂ρ(x, t) ∂ρ(x, t) ∂v(x) + · v(x) + ρ(x, t) · = 0 (8) To understand how the statement of deterministic evo- ∂t ∂x ∂x lution, or, equivalently, information conservation, can where the first two terms are equal to the total time be represented mathematically, consider first, as an ana- derivative of the density, i.e. log, the incompressible fluid flow of identical molecules in one and two spatial dimensions. dρ(x, t) ∂ρ(x, t) ∂ρ(x, t) ∂x = + · (9) For a fluid flow in one spatial dimension x, see figure dt ∂t ∂x ∂t 4, where the fluid molecules are represented as dots, the The continuity equation can thus be rewritten as velocity v of the flow is determined by the number of dρ(x, t) ∂v(x) molecules N that pass through a given location along + ρ(x, t) · = 0 (10) x during a given time interval ∆t. The rate of flow of dt ∂x Thus, if the velocity of the flow is independent on x, i.e. if ∂v(x) = 0 (11) ∂x then the density of molecules is constant in time as the fluid flow along x, i.e. dρ(x, t) = 0 (12) Fig. 4 Fluid flow in one spatial dimension. dt Such a flow is referred to as an incompressible flow be- the fluid, per time interval ∆t, is given by cause the condition that the density of molecules at any given location x within ∆x do not change over rate of flow = ρ(x)v(x) (1) time ensure that the molecules do not lump together. Thus, in conclusion, a necessary and sufficient condi- where ρ(x) is the density of molecules along x. In or- tion for the one-dimensional fluid to be incompressible der to avoid an increase or decrease in the number of is that the divergence of the flow velocity vanish, i.e. molecules within a region ∆x during an instant of time that ∂v(x) = 0. ∆t, i.e. ∂x Consider now the flow of a fluid in two spatial di- ∆N = 0 (2) mensions, x and y, see figure 5. If the number of molecules ∆t within the fixed area ∆x · ∆y within any given time in- the necessary condition is that the incoming and out- terval ∆t is to stay constant, the necessary condition going flows are equal, i.e. relating the incoming and outgoing flows is given by

ρ(xout)v(xout) = ρ(xin)v(xin) (3) ∆ (ρ(x, y)vx(x, y)) + ∆ (ρ(x, y)vy(x, y)) = 0 (13) On the Gibbs-Liouville theorem in classical mechanics 5

then the density of molecules is constant in time as the fluid flow in x and y, i.e. dρ = 0 (24) dt A necessary and sufficient condition that the fluid flow is incompressible is thus that the divergence of the flow velocity vanish.

5 The Gibbs-Liouville theorem

Fig. 5 Fluid flow in two spatial dimensions. Consider an arbitrary region Ω on a 2-dimensional phase space, with volume VΩ and volume element ∆q∆p, see figure 6. In order for the number of states N within the where ∆ (ρ(x, y)vx(x, y)) and ∆ (ρ(x, y)vy(x, y)) are defined by, respectively,

{ρ(xout, y)vx(xout, y) − ρ(xin, y)vx(xin, y)}· ∆y (14) and

{ρ(x, yout)vy(x, yout) − ρ(x, yin)vy(x, yin)}· ∆x (15) In differential form, they read ∂ d (ρ(x, y)v (x, y)) = (ρ(x, y)v (x, y)) ∆x∆y (16) x ∂x x ∂ d (ρ(x, y)v (x, y)) = (ρ(x, y)v (x, y)) ∆y∆x (17) y ∂y y Thus, the condition become Fig. 6 Hamiltonian flow on phase space. ∂ ∂ (ρ(x, y)v (x, y)) + (ρ(x, y)v (x, y)) = 0 (18) ∂x x ∂y y phase space volume Ω to neither increase nor decrease For the number of molecules to be constant over an ar- within the time interval ∆t, i.e. bitrary length of time, the necessary condition take the ∆N form, dropping spacetime coordinates in the notation = 0 (25) for convenience, ∆t it is necessary that the incoming and outgoing flows ∂ρ ∂ ∂ + (ρv ) + (ρv ) = 0 (19) cancel, i.e. that ∂t ∂x x ∂y y ∆ (ρ(q, p)q ˙) + ∆ (ρ(q, p)p ˙) = 0 (26) This is the continuity equation for the flow of molecules in two dimensions. Using the product rule and noting where ρ(q, p) is the density of states on phase space, that the total time derivative of the density is given by and the flow differences are defined by, respectively, dρ ∂ρ ∂ρ ∂ρ = + · v + · v (20) ∆ (ρ(q, p)q ˙) ≡ {ρ(qout, p)q ˙out − ρ(qin, p)q ˙in} ∆p (27) dt ∂t ∂x x ∂y y and the continuity equation can be rewritten as ∆ (ρ(q, p)p ˙) ≡ {ρ(q, p )p ˙ − ρ(q, p )p ˙ } ∆q (28) dρ ∂v ∂v out out in in + ρ · x + y = 0 (21) dt ∂x ∂y In differential form the condition 25 read ∂ ∂ or, in vector notation, (ρ(q, p)q ˙) + (ρ(q, p)p ˙) = 0 (29) ∂q ∂p dρ + ρ∇ · v = 0 (22) where dt ∂ Thus, if the divergence of the flow velocity vanish, i.e. d (ρ(q, p)q ˙) = (ρ(q, p)q ˙) ∆q∆p (30) if ∂q ∂ d (ρ(q, p)p ˙) = (ρ(q, p)p ˙) ∆p∆q (31) ∇ · v = 0 (23) ∂p 6 Andreas Henriksson

Extending the condition 25 to be valid for an arbi- then, by the continuity equation, the density of states trary length of time, the differential condition 29 be- on phase space is constant in time along the flow on come, dropping reference to the phase space degrees of phase space, i.e. freedom in the arguments of the density function for dρ convenience, = 0 (45) dt ∂ρ ∂ ∂ + (ρq˙) + (ρp˙) = 0 (32) In such a situation, the flow of the system on phase ∂t ∂q ∂p space is incompressible because the condition that the or, in vector notation, density of states at any given location (q, p) on phase ∂ρ + ∇ · (ρv) = 0 (33) space, within an arbitrary region Ω, do not change over ∂t time ensure that the states do not lump together. In where other words, in conclusion, a necessary and sufficient ∂ ∂ condition for the flow of the system on phase space ∇ ≡ , (34) ∂q ∂p to evolve deterministically, or, equivalently, to conserve is the differential operator on phase space, and information, is that the divergence of the phase flow velocity vanish. This conclusion is referred to as the v ≡ (q, ˙ p˙) (35) Gibbs-Liouville theorem [9][10]. is the velocity by which states flow on phase space. The The Gibbs-Liouville continuity equation can be de- continuity equation 33 is the Gibbs-Liouville equation rived more shortly by considering the relation between [9][10] for the density of states on phase space. It say the number of states N and the density of states ρ(q, p, t), that the number of states is locally conserved. The term i.e. ∇ · (ρv) represent the net flow of states through Ω, i.e. Z the difference between the outflow and inflow of states. N = ρ(q, p, t)dqdp (46) The continuity equation thus state that if there is a net VΩ outflow of states, i.e. if Thus, ∇ · (ρv) > 0 (36) dN d Z = ρ(q, p, t)dqdp dt dt then the density of states within Ω decrease with time, VΩ i.e. Z dρ = + ρ ∇ · v dqdp ∂ρ dt = −∇ · (ρv) < 0 (37) VΩ ∂t = 0 (47) If there is a net inflow of states, i.e. if Since information should be conserved independently ∇ · (ρv) < 0 (38) on the size of Ω, the integrand in equation 47 must then the density of states within Ω increase with time, vanish for arbitrary volumes VΩ, giving the desired re- i.e. sult ∂ρ dρ = −∇ · (ρv) > 0 (39) + ρ ∇ · v = 0 (48) ∂t dt Using that the total time derivative of the density The 2-dimensional Gibbs-Liouville theorem straight- of states is given by forwardly generalize to 6N-dimensional phase space. dρ ∂ρ ∂ρ ∂ρ = + q˙ + p˙ (40) Each conjugate pair (qj, pj), where j ∈ [1, 3N], give rise dt ∂t ∂q ∂p to an independent Gibbs-Liouville continuity equation, ∂ρ = + (∇ρ) · v (41) i.e. ∂t dρj and the product rule + ρj ∇ · vj = 0, j ∈ [1, 3N] (49) dt ∇ · (ρv) = (∇ρ) · v + ρ∇ · v (42) where ρj ≡ ρ(qj, pj) is the density of states in the 2- the continuity equation can be rewritten as dimensional subset (qj, pj) of the 6N-dimensional phase dρ space and vj ≡ (q ˙j, p˙j) is the phase flow velocity along + ρ ∇ · v = 0 (43) dt this subset. Thus, information is conserved on the 6N- Thus, if the divergence of the phase flow velocity dimensional phase space if the divergence of each phase vanish, i.e. if flow velocity vj vanish, i.e. if

∇ · v = 0 (44) ∇ · vj = 0 ∀j ∈ [1, 3N] (50) On the Gibbs-Liouville theorem in classical mechanics 7

6 Hamilton’s equations

The vanishing divergence of the flow velocity vj for all conjugate pairs (qj, pj), j ∈ [1, 3N], written out explicitly in terms of its velocity componentsq ˙j andp ˙j, become ∂q˙ ∂p˙ j + j = 0 ∀j ∈ [1, 3N] (51) ∂qj ∂pj Let H be a smooth function on the 6N-dimensional phase space with the property that it contain no terms that mix different conjugate pairs, e.g. pi · pj, ∀i 6= j. In this situation, taking into account that the set of 3N Fig. 7 The areas underq ˙j (pj ) andp ˙j (qj ) graphs define the conjugate pairs {(qj, pj)}j=1 are postulated to be in- Hamiltonian and Lagrangian, respectively. dependent, the condition of vanishing divergence can equivalently be stated by the set of differential equations known as Hamilton’s equations, This integral equation correspond to the differential equation1 ∂H q˙j = ∀j ∈ [1, 3N] (52) dL(q ˙j) ∂pj = pj (56) dq˙j ∂H p˙j = − ∀j ∈ [1, 3N] (53) The total area of the rectangle bounded by (0, pj) and ∂qj (0, q˙j) is given by Under these circumstances, the Hamilton equations are, L(q ˙ ) + H(p ) = p · q˙ (57) according to the Liouville-Arnold theorem [11][12], in- j j j j tegrable from a set of known initial conditions. This It is possible to include a dependence on the generalized simply mean that they describe an evolution of the sys- coordinate qj under the constraint that any qj−dependent tem which is unique and deterministic. Thus, given the terms in the functions H and L cancel such that the to- function H, the flow of the system in time is determined tal area is qj−independent. Thus, in general, the func- by how H change on phase space. In this sense, H is tions H and L, satisfy the so-called Legendre transfor- said to be the generator for the motion in time of the mation, system on phase space. The flow of the system on phase L(qj, q˙j) + H(qj, pj) = pj · q˙j (58) space, described by the Hamilton equations, is referred to as an Hamiltonian flow. where Z q˙j L(qj, q˙j) = dq˙j pj(q ˙j) − U(qj) (59) 0 Z pj 7 The Hamiltonian and Lagrangian H(qj, pj) = dpj q˙j(pj) + U(qj) (60) 0

The Hamilton equation 52, for a specific conjugate pair The requirement that the total area is qj−independent (qj, pj), correspond to the integral equation cause the Lagrangian and Hamiltonian to have a rela- Z tive sign difference for the function U(qj). H(pj) = dpj q˙j(pj) (54) For the 6N-dimensional phase space, the functions H and L, referred to as the Hamiltonian and Lagrangian, The momentum pj and speedq ˙j are assumed to be respectively, are defined by in one-to-one correspondence. This mean that for each 3N Z q˙j value ofq ˙ there is a unique value for p , and vice versa. X j j L(q, q˙) ≡ dq˙j pj(q ˙j) − U(q) (61) The function H(pj) is then geometrically interpreted as j=1 0 the unique area under theq ˙j(pj)−graph, bounded by 3N X Z pj (0, pj) and (0, q˙j(pj)), see figure 7. Due to the one-to- H(q, p) ≡ dpj q˙j(pj) + U(q) (62) one correspondence between pj andq ˙j it is possible to j=1 0 define a related area, L(q ˙j), given by the unique area (63) under the pj(q ˙j)−graph, 1 In the Lagrangian formulation of classical mechanics, this Z differential equation is the defining equation for the momenta L(q ˙j) = dq˙j pj(q ˙j) (55) conjugate to the generalized coordinate. 8 Andreas Henriksson where the function U(q), defined by Integration by parts and recalling the boundary condi- 3N tions give X U(q) ≡ U(qj) (64) Z tf ∂(q ˙jpj − H) d ∂(q ˙jpj − H) j=1 0 = dt − δqj(t) t ∂qj dt ∂q˙j is referred to as the potential energy of the system. i Z tf ∂(q ˙jpj − H) d ∂(q ˙jpj − H) + dt − δpj(t) ti ∂pj dt ∂p˙j 8 Principle of stationary action Z tf = dt δ (q ˙jpj − H) The Hamilton equations ti Z tf ∂H = δ dt (q ˙jpj − H) − − p˙j = 0 (65) ti ∂qj Z tf ∂H = δ dt L(qj, q˙j) q˙j − = 0 (66) t ∂pj i = δA(q , q˙ ) (74) is the local, differential, representation of the principle j j of information conservation on phase space. A global, where or integral, representation can be obtained by consider- Z tf ing the entire evolutionary path from some initial time A(qj, q˙j) ≡ dt L(qj, q˙j) (75) ti ti to some final time tf where the Hamilton equations 2 are integrated over time . For this purpose, multiply is the action of the system within the subset (qj, pj) on the Hamilton equations with two independent arbitrary the 6N−dimensional phase space. The action on the functions of time, δqj(t) and δpj(t), representing, re- entire phase space is given by spectively, small displacements in qj and pj on phase 3N Z tf space, in the following manner, X A(q, q˙) ≡ dt L(qj, q˙j) ∂H j=1 ti − − p˙ δq (t) = 0 (67) j j tf ∂qj Z = dt L(q, q˙) (76) ∂H ti q˙j − δpj(t) = 0 (68) ∂pj This is Hamilton’s formulation of the principle of sta- The displacements δqj(t) and δpj(t) are pictured as tionary action, or shortly, Hamilton’s principle. It is a slight variations of the physical path on phase space, global representation of information conservation, i.e. a i.e. statement on the entire evolutionary path which must be satisfied if the system is to adhere to the principle qj(t) → qj(t) + δqj(t) (69) of information conservation. pj(t) → pj(t) + δpj(t) (70) Since the Hamilton principle can be derived from Equations 67 and 68 are equivalent to the Hamilton the Hamilton equations, which in turn is an immediate equations since they hold for arbitrary variations. The consequence of the requirement that the divergence of fact that it is necessary to introduce two displacement the Hamiltonian flow velocity vanish, it should be pos- functions is due to the independence of the state pa- sible to obtain the Hamilton principle directly from the rameters qj and pj. The boundary conditions are given requirement that ∇ · vj = 0 is invariant under the dis- by placements δqj(t) and δpj(t). Given that the variations are small, the flow velocity vj can be expanded as a δqj(ti) = δqj(tf ) = 0 (71) Taylor series about the state (qj, pj) where terms that δpj(ti) = δpj(tf ) = 0 (72) are of quadratic, or higher, order in the variations δqj i.e. the variations vanish at the initial and final times. and δpj can be ignored. The infinitesimal change in vj Integrating the Hamilton equations over time from ti thus become to tf give, to leading order in the variations, δvj = vj(qj + δqj, pj + δpj) − vj(qj, pj) Z tf ∂H ∂H dt − − p˙ δq (t) + q˙ − δp (t) = 0(73) ∂ ∂ ∂q j j j ∂p j = δqj vj + δpj vj (77) ti j j ∂qj ∂pj 2 For the derivation of an integral representation on config- The divergence of the flow velocity transform as uration space starting from Newton’s second law of motion, see chapter 10 in reference [13]. ∇ · vj → ∇ · (vj + δvj) = ∇ · vj + ∇ · δvj (78) On the Gibbs-Liouville theorem in classical mechanics 9

If ∇ · δvj 6= 0, information is not conserved for the The equality must hold independently on the surface deviated path. Therefore, it is required that area of the tube, i.e. the principle of information conservation should hold true independently on the number ∇ · δvj = 0 (79) of states in which the system can exist. Therefore, the which is equivalent to integration over the surface area can be taken outside

δ (∇ · vj) = 0 (80) of the infinitesimal variation, giving that This statement is for a blob of volume dV which en- Z tf δ dt L = 0 (89) close the single state (qj, pj). Information conservation ti should hold for all varied states along the evolutionary which is, again, Hamilton’s principle. path of the system, from the initial state (qj, pj)i, at time ti, to the final state (qj, pj)f , at time tf . Thus, the above statement should be integrated over all blobs 9 Invariance of the Poisson algebra of volume dV along the path, i.e. the integration is over a tube, with volume V , whose interior define the region Given that the divergence of the Hamiltonian flow ve- of extended phase space where the principle of informa- locity vanish, the Gibbs-Liouville equation can be writ- tion conservation is fulfilled. Thus, ten as

Z tf Z ∂ρ + ∇ρ · v = 0 (90) δ dt dV ∇ · vj = 0 (81) ∂t ti V Applying the divergence theorem The Poisson bracket {ρ, H} between the density of states Z Z ρ and the Hamiltonian H is defined by dV ∇ · vj = dS · vj (82) ∂ρ ∂H ∂ρ ∂H V ∂V {ρ, H} ≡ ∇ρ · v = − (91) ∂q ∂p ∂p ∂q give In general, the Poisson bracket {A, B} between any two Z tf Z δ dt dS · vj = 0 (83) arbitrary functions A and B on phase space is defined ti ∂V by The integrand dS · vj represent the density of the net ∂A ∂B ∂A ∂B Hamiltonian flow out of the tube. The surface area el- {A, B} ≡ − (92) ∂q ∂p ∂p ∂q ement dS is given by In this notation, the Hamilton’s equations are written dS = dS n (84) as where n = (p , q ) is the normal vector to the surface j j q˙ = {q, H} (93) of the tube, i.e. n give the direction in phase space in which the system has to flow if it is to eventually p˙ = {p, H} (94) reach a region where the principle of conservation of The Poisson bracket satisfy a set of algebraic properties. information no longer hold. Thus, with vj = (q ˙j, p˙j), It is antisymmetric, i.e. the integrand become {A, B} = − {B,A} (95) (pj, qj) · (q ˙j, p˙j) = pjq˙j + qjp˙j (85) It satisfy linearity, i.e. R Using that qj = dqj and the Hamilton equationp ˙j = − ∂H , the integrand can be written as {aA + bB, C} = a {A, C} + b {B,C} (96) ∂qj Z ∂H Z Furthermore, it satisfy the product rule and the Jacobi pjq˙j − dqj = pjq˙j − dH = pjq˙j − H (86) identity, i.e. ∂qj Equivalently, the integrand could have been written as {AB, C} = A {B,C} + {A, C} B (97) qjp˙j + H (87) {A, {B,C}} + {B, {C,A}} + {C, {A, B}} = 0 (98) R by using that pj = dpj and the other Hamilton equa- These properties define the Poisson algebra of classical ∂H tionq ˙j = . However, the form pjq˙j − H is the pre- ∂pj mechanics. Since the Gibbs-Liouville equation for the ferred choice due to the fact that it is equal to the La- incompressible Hamiltonian flow can be expressed in grangian L(qj, q˙j). Thus, on the 6N-dimensional phase terms of the Poisson bracket, the Gibbs-Liouville theo- space it is obtained that rem can equivalently be stated by saying that the evo- Z tf Z lution in time of any given system is deterministic if it δ dt dS L = 0 (88) leave the Poisson algebra invariant. ti 10 Andreas Henriksson

10 Unitary time-evolution operator 5. Lagrange, J.L.: Mecanique Analytique, vol. I. M.V. Courcier, 1st ed., Paris (1811) Since the ordering of ∇ρ and v do not matter in their 6. Lagrange, J.L.: Mecanique Analytique, vol. II . Mallet- Bachelier, 3rd ed. , Paris (1815) scalar product, the Gibbs-Liouville equation can equiv- 7. Hamilton, W.R.: On the application to dynamics of a alently be written as, after multiplication with the imag- general mathematical method previously applied to op- inary unit i, tics. British Association Report, pp. 513-518. Edinburgh (1834) ∂ρ i − Lρ = 0 (99) 8. Moyal, J.E.: Quantum mechanics as a statistical theory. ∂t Proceedings of the Cambridge Philosophical Society, 45, p.99-124 (1949). where the differential operator L is referred to as the 9. Gibbs, J.W.: Elementary principles in statistical mechan- Liouville operator and defined by ics. Charles Scribner’s sons, New York (1902) 10. Bloch, F.: Fundamentals of statistical mechanics: L ≡ −iv · ∇ (100) manuscript and notes of Felix Bloch. Imperial College † Press and World Scientific Publishing, 3rd ed., London The Liouville operator is self-adjoint, i.e. L = L, with (2000) the consequence that its eigenvalues are real-valued [14]. 11. Liouville, J.: Note sur l’intégration des équations The formal solution to the Gibbs-Liouville equation, in différentielles de la Dynamique. Journal de terms of the Liouville operator, is mathématiques pures et appliquées 1re série, tome 20, p. 137-138. (1855) ρ(q, p, t) = e−iL·tρ(q, p, 0) (101) 12. Arnold, V.I.: Mathematical methods of classical mechanics. Springer-Verlag, 2nd ed. (1989) The operator U(t), defined by 13. Jeffreys, H., Jeffreys, B.S.: Methods of mathematical physics. Cambridge University Press, 3rd ed. (1956) U(t) ≡ e−iL·t (102) 14. Arfken, G.B., Weber, H.J.: Mathematical methods for physicists. Harcourt Academic Press, 5th ed. (2001) generate the flow in time of the density of states and is therefore referred to as the time-evolution operator. It is unitary, i.e. U †U = 1 (103) The connection between the notion of unitary evolution and determinism is thus as follows. If a given system evolve unitarily in time, described by the time-evolution operator U(t), then it is guaranteed that the divergence of the Hamiltonian flow velocity vanish. Thus, the Gibbs-Liouville theorem can equivalently be stated by saying that the evolution in time of any given system is deterministic if it is described by the unitary time-evolution operator U(t).

Acknowledgment

The author would like to express gratitude to Anke Olsen for many inspiring discussions in our oﬃce at Bacchus.

References

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