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Ten1perature, Least Action, and Lagrangian Mechanics L 14 Auguq Il)<)5 2) 291 PHYSICS LETTERS A 's. Rev. ELSEVIER Physics Letters A 204 (1995) 133-135 l) 759. 7. to be Ten1perature, least action, and Lagrangian mechanics l. 843. William G. Hoover Department ofApplied Science, Ulliuersity ofCalifornia at Davis / Livermore alld Lawrence Licermore Nalional Laboratory, LiueTlrJOre, CA 94551-7808, USA Received 28 March 1995; revised manuscript received 2 June 1995; accepted for publication 2 June J995 Communicated by A.R. Bishop Abstract Temperature is considered from two different mechanical standpoints. The more approach uses Hamilton's least action principle. From it we derive thermostatting forces identical to those found using Gauss' Principle. This approach applies to equilibrium and nonequilibrium systems. The Lagrangian approach is different, and less useful, applying only at equilibrium. Keywords: Least action principle; Nonequilibrium; Lagrangian mechanics 1. Introduction When the context is clear we use just a single q or q to represent whole sets. Feynman repeatedly emphasized the generality of Lagrangian mechanics is particularly well-suited Hamilton's principle of least action [1], the physical to the treatment of constrained systems. Constraints, law from which both kinds of mechanics, classical used first to represent mechanical linkages and cou­ and quantum, follow. The principle states that the plings, have more recently been used in molecular observed trajectory {qed} is that which minimizes simulations, where they maintain the shape of a the action integral, polyatomic molecule and reduce the number of dy­ namical equations to be integrated. The usual ap­ ofL(q,q,t)dt of(K <P)dt=O, proach is to add a "Lagrange multiplier" to the Lagrangian in such a way as to impose the constraint where K is the kinetic energy, <P is the potential without affecting energy conservation [2,3]. For in­ energy, and L(q, q) K q) is the Lagrangian. stance, a particle with unit mass, constrained to The integration time interval is fixed, as are also the circle the origin, at unit radius, can be described with coordinate sets at both end points. Here we restrict the Lagrangian our attention to equilibrium or nonequilibrium many-body systems with kinetic energies quadratic L(x,y,x,y) in the velocities and potential energies which depend only on the coordinates {q} In this case, the Lagrangian equations of motion, K= Lmq2j2, <P <P(q). (djdt)(aLjaq)q (aLjaq)q, 0375-960] /95/$09.50 1995 Elsevier Science B.Y. All rights reserved SSDI 0375-960 1(95)00464-S 1:\4 WG. /lOOl'I'f / {,hysics LeU!'fS A 204 (iCj')5) /.!3-/35 multiplier? to insure that the solution is consistent with the variation of the isokinetic constraint, x = 2Ax, y 2Ay. oQo(a/aqo)(m/2 d(2)[ (qo q _)2 - (q.! qu)2] The value of the Lagrange multiplier A follows from the second derivative of the constraint O. Adding the two variational equations, and then tak­ ing the limit dt -7 0 gives the isokinetic differential It is unnecessary to specify whether the Lagrange equation of motion, multiplier depends upon time, upon space, or even (la) space, velocity, and time simultaneously, because the corresponding derivatives in the equations of motion The Lagrange multiplier ? can be identified, as in would all be multiplied by the constraint 1, and the simple rigid-rotor example, by evaluating the therefore vanish automatically. time derivative of the constraint. The result for the In the next two sections we apply the two ideas Lagrange multiplier!: (which in this case has the just described, Hamilton's principle and Lagrangian form of a "friction coefficient") is constraints, to the more complicated case of con­ (lb) straining temperature(s) in many-body systems [3,4]. As is usual in classical nonequilibrium statistical and provides familiar time-reversible constraint mechanics, we define temperature [5] in terms of the forces. This result was already known to follow from mean-squared kinetic energy of a typical Cartesian Gauss' principle of least constraint [8,9], but Hamil­ degree of freedom, ton's least action principle is more general, because it promises also the possibility of extending the kT=. < ). thermostat idea to nonequilibrium quantum systems. In the following section we will see that a straight­ forward application of classical Lagrangian mechan­ ics to the same problem yields a less useful result. 2. Temperature and Hamilton's principle Let us apply the least action principle to trajecto­ 3. Temperature and Lagrangian mechanics ries defined in the infinitesimal time interval be­ tween dt and +dl, considering all the coordinates If we choose to use a Lagrange multiplier, A, to be fixed at the endpoints, and with the set of which varies with time, and can also depend upon coordinates at the central time, {qo}, able to vary [6]. the coordinates and velocities, and use it to impose The centered-difference approximation to Hamilton's the constant-kinetic-energy constraint, principle, L( q, q) d t = 0, of the resulting equation of motion is is (d/dt)[mq(l+A)] F(q) 2 mij(l + A) + Amq F(q). oqo(a/aqo) {( m/2 dt ) [( qo q)2 + (q + - qO)2] = Multiplying by it, and summing over degrees of H<P _+ 2<1\ + <P+)} dt freedom, gives a simple differential equation for .A, oqo[(m/dt2 )(2qo-q_-q+) +FoJ dt=O, A.'[mq2 = 2A. Ko F'q~ -$ E)/2K , giving the usual StOrmer-Verlet equation of motion = .A=( o [6,7]. To add temperature we introduce a Lagrange where Eo is the energy at the initial time, t o. l\' NOlII'('/' / Physics LCI/ers:\ 2(N (191)5) /33-/35 135 istent Notice that the {illle deril'atil'<' of the Lagrange coefficients, {{I}' These same ideas apply to isobaric multiplier, A, is <:xactly equal to the friction coeffi­ [12,13] or isoenergetic nonequilibrium algorithms. cient ( found previously using the principle of leasl Because the least action principle applies to quantum action. To make this identification explicit, we write systems too, there is potential for developing new the present corresponding time-reversible Lagrangian methods for treating nonequilibrium quantum sys­ equa~ion of motion in terms of both these multipliers, tems [14]. Our specific isokinetic example illustrates tak­ ( A and A, the fundamental limitation of Lagrangian mechanics ;nlia1 mq=[F(q) (mq]/(1 A), to equilibrium. (1a) (= LJ·q/E2Ko. Acknowledgement as in The equation of 1110tion has an interesting form. For :., the small fluctuations in the energy, as for instance occur I thank Brad Holian (Los Alamos), Dave Boer­ I~ the in a large equilibrium system, the Lagrange multi­ cker, Carol Hoover and Oyeon Kum (Livermore), is the plier A is small, and the motion reduces to the and Harald Posch (Vienna) for many useful conver­ Gaussian form (1) found above. On the other hand, sations concerning the development of algorithms for in a nonequilibrium steady-state system, the inte­ nonequilibrium simulations. Support of the staff at grated energy change (which is mainly heat extracted the Habitat (Netherlands Antilles), where a part of by the thermos tatting constraint forces), and the La­ '[raint this work was completed, is gratefully acknowl­ from grange multiplier A, grow without bound, so that the edged, as is also support at the Lawrence Livermore ami!­ equations have no long-time solution. National Laboratory under United States Department cause Thus the Lagrange-multiplier approach, unlike the of Energy Contract W-7405-Eng-48. " the principle of least action, is not applicable to ~ems. nonequilibrium systems. This finding confirms Feyn­ ight­ man's emphasis of the fundamental nature of least References han­ action. Evidently classical Lagrangian mechanics is ult. restricted to equilibrium problems in a way which [1] R. Feynrnan, The character of physical law (Modern Library, the principle of least action is not. New York, 1994). [2] G. Ciccotti and J.P. Ryckaert, Comput. Phys. Rep. 4 (1986) 345. 4. Summary [3] W.G. Hoover, Computational statistical mechanics (Elsevier, Amsterdam, 1991). r, A, [4] S. Nose, Prog. Theo. Phys. SUpD!. 103 (1991) 1. ,upon We have here generalized the basis of Gaussian [5] W.G. Hoover, RL. Holian and HA Posch, Phys. Rev. A 48 lpose isothermal mechanics by deriving the equations of (1993) 3196. motion directly from Hamilton's least action princi­ [6] R.E. Gillilan and KR. Wilson, J. Chern. Phys. 97 (1992) ple. This approach yields algorithms useful in simu­ 1757. [7] D. Levesque and L. Verict, 1. Stat. Phys. 72 (1993) 519. lating nonequilibrium systems too, by restricting the [8] W.G. Hoover, AJ.e. Ladd and R Moran, Phys. Rev. Lett. number of degrees of freedom to be constrained. For 48 (J 982) 1818. very recent summaries of nonequilibrium work see [9] DJ. Evans. W.G. Hoover, RH. Failor, B. Moran and A.J.e. Refs. [lO,ltJ. In the simple case of nonequilibrium Ladd, Phys. Rev. A 28 (1983) 1016. many-body heat flow, a set of bulk particles obeys [10] W.G. Hoover, Physica A 194 (1993) 450. [11] E.G.D. Cohen, Physica A 213 (1995) 293. the usual Newton equations, while two reservoirs, [12] S. Nose, J. Chern. Phys. 81 (1984) 511. of es one hot and one cold, obey the constrained Gauss [13] W.G. Hoover, Phys. Rev. A 34 (1986) 2499. lr A, equations of motion (1) with two separate friction [14] D. Kusnezov, Phys. Lett. A 184 (1993) 50. ). .
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