Pictorial Representation of Irreversible Processes Driving Particles Toward Equilibrium B. Boulil, 0. Hemi-R-au Universite de Perpignan, 66025 Perpignan Cedex. France P. Blaise Universite de ~alenciennes.59326 Valenciennes Cedex, France Systems at thermodynamical equilibrium that are timeindevendent are eenerallv studied before dealine with nonequiiibrium situations. ~ i t h o u the ~ h time development of a noneauilibrium situation is harder to describe mathematically,the study of such systems provides a deeper understanding of the equilibrium state. In statistical mechanics the time evolution of a nonequilibrium situation is governed by the Fokker-Planck equation ( I ) in the framework of the Markoff approximation. Its solutions permit a pictorial representation of how molecular systems evolve toward equilibrium as a result of irreversible processes (2). Unfortunately, the textbooks in physical chemistry do not make reference to this fundamental equation nor to graphical representations of its solutions (3). Our purpose is to emphasize the importance of the Fokker-Planck equation in assisting an intuitive comprehension of the basis of equilibria. We shall start from the FokkerPlanck equation considered as a principle. Then, we shall discuss its solutions and give pictorial descriptions of the path of irreversible processes.
where k is the Boltzmann constant and n is a coefficient connected to the viscosity s of the thermal bath by Stokes's law: a = 6rqr
The solution of eq 1is
Equation 3 reduces as required for t distribution, since we may then write (5)
The Brownlan Particle in a Thermal Bath (4)
Consider the movement of a macroscopic particle interacting with a surrounding medium at any temperature, that is, a Brownian particle. First, suppose we look at any given 'varticle, the velocitv of which was known to be un at some initial time to. As time goes on, the velocity will bemodified abruptly from time to time becauseof the collisions with the molecules of the surroundings. Thus, the velocity of the Brownian particle evolves in a stochastic way (5). Now, if we are dealing with a large collection of particles in place of a single one, but all starting at time to with the same velocity uo, it is possible to look at the local average behavior of these particles as they evolve from the same initial state. Then, as time aoes on. these articles become on the average more and &re like the mdlecules of the fluid with which ;hey are interacting, so that for infinite time. the Brownian varticles must come to equilibrium with the surroundings medium regardless of their past history. The Time Evolution of the Dlstrlbution Governing the Brownlan Particle For this purpose, let us ask a detailed question concerning the time-dependent probability P(u,, uo., t)du,. Does the x component of the particle velocity at time t lie between u. and u, du, if u, = uo, at time to? We shall denote by m and r the mass and the radius, respectively, of the particle, which we shall consider as spherical, and by 7 and T, the viscosity and the temperature of the surrounding medium acting as a thermal bath. The answer to this question may be obtained from statistical mechanics which gives for P(u,, u b , t ) the following partial differential equation, known as the Fokker-Planck equation ( I ) . Journal of Chemical Education
0 to a Dirac 6
-
where r 0. On the other hand, for infinite time, eq 3 reduces to the Maxwell distribution:
The Time Evolution of the Brownlan Partlcle With the help of the time-dependent distribution given by eq 3, the time evolution of the first and second moments of the velocity of the Brownian particle under consideration are, respectively, the following:
Then, using eq 3, we obtain by integration:
+
714
-
u ,=~ u,
exp
[z]
(8)
and
Next, we may observe that for infinite time, these average values reduce to u,(*) = 0
(10)
V"
butions toward the same equilihrium value. Meanwhile, the half width of each distribution is increasing promessively so as to reach that of an&ifibrium Boltzmann distribution for infinite time. Figure 1may he viewed as an exhibit of irreversible processes that are connected to the stochastic collisions of the Brownian narticles. This figure is an illustration that the irreversible processes connected to the stochastic collisions of the Brownian particles all drive toward the same final state with a mogressive lack of the memory of the past, regardless of the initial conditions. Consequently, Figure 1 may he considered as a pictorial representation of the "arrow of time". Pictorial Representation of the Temperature Effect
As is well known, raising the temperature leads LO an increase in the disorder of a svstem and thus to abroadening of the distribution. This effect is taken into account by eq 3, as may be observed by inspection of Figures 2a and 2h where the time behavior of two identical Brownian particles is compared. The two particles start from the same initial conditions and evolve in a medium of the same viscosity but at different temperatures.
Figwe 1. m e dlabibutions P(v, fi, 1) versus tlme. hom two distlnn 6 Dirac at time f = 0 to me ~ w e Idisblbution I for intinhe time and me lack of m e m w of b e initial mnditioos Whatthe initialvalue of v,isattlme t = 0, thedisblbutionscoalesceprogressivelytowardthe same Maxwell dlsfrlbution fw Infinitetime.
and
which are the usual results (6) obtained by averaging respectively u, and uz2on the Maxwell distribution (5):
Pictorla1 Representatlon of the Viscosity Effect
In order to visualize the viscosity effect on the relaxation process, we compare two Brownian particles of the same
and
Moreover, by aid of the moments of u,, as given by eqs 8 and 9, we may find the dispersion Au,(t) on the velocity, which is given according to A"#)
- = [u,2(t) - ~ ~ ( t ) ~ ] ~ ' ~
(14)
Performing the calculation, we get
a result that for infinite time reduces to the usual one (7):
that we may obtain directly by aid of eqs 10,11, and 14. Pictorlai Description of the Tlme Evolution of the Distribution
For most of our purposes, i t is sufficient to limit the presentation of the results of our treatment to the pictorial description of the time evolution of the distribution P ( t ) as given by eq 3. This description is in a three-dimensional representation where in the first dimension we have the velocity u, of the particle, in the second we have the time t , and in the last one, the prohahility P ( t ) . Figure 1gives the change with time of two distrihutions starting at time t = 0 from two distinct Dirac 6 distrihutions and evolving with time. As can be seen, the positions of the maxima of each distribution are moving in a continuous way from their initial values corresponding to the Dirac 6 distri-
Figure 2. The temperature effect on me time evolution of the distributions: (a) T = 400 K: (b) T = 4000 K.
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nature, starting from different initial situations and evolving in two media a t the same temperature but different in their viscosity coefficients. In Figure 3a, 00, and cl are both larger than in Figure 3b. As can he seen, the second distribution is approaching equilibrium faster, corresponding to the greater viscosity coefficient.
volved, statistical mechanics leads to the following differential equation governing the conditional probability P(x, xo,
Time Evolution ot the Posnion Probablllly of a Partlcie
where B is a constant characterizing the surroundings. The nature of the B constant appears clearly in the special case where X = 0, that is, when we consider the time evolution of the average position of a Brownian particle. Then, eq 17 reduces to
Now let us consider a macroscopic system, submitted to the restoring force obeying Hookes's law and interacting with a surroundine medium. and ask the detailed auestion concerning the tige dependence of P(x, xo, t). How does one find the x component of the particle lying between x and x dx at time t, if at time to it &as xo? &&a system may be a quasi-classical harmonic oscillator (8.9) embedded in a viscous medium. Recall that such a quasi-classical system is described by a state that is a linear combination of eigenstates of the Hamiltonian of the quantum harmonic oscillator for which the effect of the Heisenberg uncertainty relation is a minimum (8). In such a state, the average values of the position, of the momentum, and of the kinetic energy oscillate back and forth with time, according to the corresponding equations of classical mechanics. Of course, because of these special properties, such a system may be damped by interaction with the medium. To make the model more concrete, the system may be a torsional pendulum embedded in a viscous medium, a cavity mode of laser (8) or the slow X - H - -- Y vibrational mode of a X-H molecule involving hydrogen bond with an Y system (10). If the spring constant A characterizing the restoring fzrce is zero, the new system we obtain is similar to that of a Brownian particle interacting with the medium and for which we may ask about the time evolution of its probability position assuming that it was defined at any initial time. In the general situation where the harmonic force is in-
t) :
+
a result that appears to be the second Fick diffusion equation (8):
where D is the diffusion constant that is connected to the temperature and to the B constant through
As we may observe, the differential eq 17 is of the same form as eq 1.Then, its solution has also the same form as that of eq 1, which is given by eq 3. In the present situation, the solu' tion is
Because of the similarity of this equation with eq 3, the graphical representation is the same as that given in the figures. Thus, these figures may he used in the present situation by changing only the comments. I t is worthwhile to remark that when A is zero, the solution of eq 21 is that of the second Fick diffusion equation:
a result that is a random walk. On the other hand, it may be of interest to verify that for infinite time eq 21 reduces to
a result that may be obtained directly from Boltzmann statistics owing.to the potential energy E. of an harmonic oscillator being
Conclusion
time
F l w e 3. The mediumviscosity enect on 6-e timeevolutionof6-edlsbibulions (a) a = 1; (b) a = 2.
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By aid of the Fokker-Planck equation, considered as a starting point, it is possible to supply undergraduate students with a pictorial representation of irreversible processes carrying an average Brownian particle from any initial situation toward the thermal distribution. Such an approach permits an intuitive understanding of how an equilibrium state is reached as the aswontotic limit in infinite time. This single thermodynamic o;tcbme is the consequence of different initial condi~ions.the memorv of which has been lost in the final state. Moreover, by aid of a computer, i t is possible to present on a video screen the time evolution of the Fokker-Planck distributions. A fh on avideotape made from such a computer presentation ought to be useful in teaching the nature i f the
internal energy a t the microscopic level a t thermal equilibriurn for the Maxwell distribution of velocities.
3. (a1 Atkina, P. W.Physical Chemhlm; Oxfwd Uniu.: Oxford, 1978 (bl Mahan, 6. H.
Uniuersity Chemhfry: A d d i m - W d e y Reading. MA. 1969. 4. Reif,?. findomentoS of Stofirticol end Thermal Physics: MeGraw-Hill: Nea. Yorr
"
1-5. . " * " , v " .Z"Z v.
Literature Cited 1. Van Kampsn, W. G. Stachmtic h e s w in Physics ond Chemhtm: North Holland: Amsterdam, 1981: p 2W.See ako M a n h e , D. A. Sfofhficol Mechanics; H w e r and Rw: New York, 1976: Chap& 14. 2. MaeDonsld, D. K. C. Noise ond Nuclmfions: An htmdwtion; Wiley: New York,
1962; D 71.
5. ~ ~ ~ D . G J.c R k m. . ~ d u r IJSI.SI,II. . 6. Barrow, G. M. Pfiysical Chemhtw;Mffiraw-Hill: Nea. York, 197%p n . 7. Moelm-Hughes. E. A. Physicel CheMa1ry;Pergamon: Oxford. 1991; p 1211. Wiley: New York, 1973; 8. LouiseU. W. H. QmntumStathlieolPm~ertiesoiRodiofion; p 334. 9. Boulil. B.; Hem-Rou-u, 0. J. Chem. Edw., in prem. 10. Boulil, 8.; H s n n - R a w u , O . ; B b c , P.Chem. Phya. 1988,126,263.
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