3061
Momentum Autocorrelation Function of a Heavy Particle in a Finite Crystal Robert J. Rubin
Contribution f r o m the National Bureau of Standards, Washington, D. C. 20234. Received Nouember 22, 1967 Abstract: The momentum autocorrelation function of a heavy particle in a finite one-dimensionalcrystal, po")(7), is compared with the same function in an infinite crystal, p 0 ( - ) ( 7 ) , by obtaining an explicit upper bound for the mag-
nitude of the difference, l f i ( N ) ( ~-) p o ( " ) ( ~ ) ] . The precise meaning of the statement that p O ( " ) ( 7 )is approximately a ) exponentially. simple exponential is reviewed, and precise meaning is given to the statement that B ( ~ ) ( T decays
0
ver 50 years ago, Professor Debye recognized that density variations or fluctuations play an essential role in the interpretation of the thermal conductivity of crystalline so1ids.I His paper was one of the first on the properties of crystals containing defects. There is now an enormous and still-growing literature on the subject. The investigations of the time-dependent properties of a heavy particle substituted in a harmonic crystal have been extensive and n u m e r o u ~ ~ -because '~ of the relevance of this system to statistical mechanical theories of Brownian motion. In these investigations, the momentum autocorrelation function of the heavy Brownian particle has been evaluated only in the limit of an infinite crystal. In the present paper we obtain estimates of the same momentum autocorrelation function in the case of a finite one-dimensional crystal. This work was stimulated by a discussion with R. W. Zwanzig, and it is based on a method used recently by Rubin and Ullersmal* for estimating the momentum autocorrelation function of particles in a special finite harmonic oscillator system, the Bernoulli chain. The one-dimensional, nearest neighbor crystal model, 1 particles with which we consider, consists of 2N periodic boundary conditions. The particles are labeled from - N to N , and all particles except 0 have a mass m while particle 0 has the mass M > m. There are two equivalent expressions for the mOmentum autocorrelation function of particle 0 in a finite crystal.
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(1) P. Debye in "Vortrage iiber die Kinetische Theorie der Materie," Teubner, Leipzig and Berlin, 1914. (2) R. J. Rubin in "Proceedings of the International Symposium on Transport Processes in Statistical Mechanics, August 1956," I. Prigogine, Ed., Interscience Publishers, Inc., New York, N . Y.,1958, p 155. (3) P. C. Hemmer, Thesis, Norges Tekniske Hogskole, Trondheim, Norway, 1959. (4) R. J . Rubin, J . Math. Phys., 1, 309 (1960). ( 5 ) R. E. Turner, Physicu, 26, 269 (1960). (6) R. J. Rubin, J . Math. Phys., 2, 373 (1961). (7) M. Toda and Y. Kogure, Progr. Theoret. PhJ)S. (Kyoto), Sirppl., 23, 157 (1962). (8) M. Toda, ibid., 23, 172 (1962). (9) S. Takeno and J. Hori, ibid., 23, 177 (1962). (10) P. Mazur in "Proceedings of the International Symposium on Statistical Mechanics and Thermodynamics, June 1964," J. Meixner, Ed., North-Holland Publishing Co., Amsterdam, 1965, p 69. (11) P. Mazur and E. Braun, Physica, 30, 1973 (1964). (12) R . J. Rubin,J. A m . Chem. Soc., 86, 3413 (1964). (13) P. Ullersma, Physica, 32, 74 (1966). (14) H . Nakazawa, Progr. Theoret. Phj,s. (Kyoto), Suppl., 36, 172 (1966). (15) H . M'ergeland, Kgl. Norske Videizskub. Selskabs, Forh., 39, 109 (1966). (16) I. Fujiwara, P. C . Hemmer, and H . Wergeland, Progr. Theoret. PhJ)S.(I>
(17) 1 and
7
6 7
7T-I''
3
&/a
and T E 18.42. Thus in this example, after 18 relaxation times, the correction to the exponential is less than of the value of the exponential. It is clear that as Q increases, the T interval in which the exponential is a a In Q and good approximation to p o ( " ) ( T ) increases the error in the approximation approaches zero @-('Iz) (In where a < 5 / 2 . Finite N , N >> Q >> 1. We now consider the inequality (19) when N >> Q >> 1. For T = a In Q, the ratio of the error bound to the exponential in eq 19 is a = ( ~ / T ) ' / ~ Q = - ( ' / ~In )(cY +
-
-
e)-'//'
+ (aT + '/-JQN-l
N-lQ1+OL(aaIn Q
(18)
+
1/2)
(21)
In the preceding case the term proportional to N-l was absent. In the present case, if Q >> 1 and N = where 0 < p < (5/2) - a, then the second term on the right-hand side dominates the first
5 2'/2Q-1X
min(l,a-'/YQT)-s/2)
a = ( 2 / ~ ) ' / 2 @ - ( ' / 2 ) In ( a e)-*/, (20) The ratio approaches zero for CY < 5 / / 2 . For example, when CY = 2 and Q = lo4, the ratio a E 1.01 X
e)-'/,
Combining (17) and (18), we obtain finally I P ~ ( ~ )( Texp(--T)/ )
bound to the exponential in eq 19 is
(19)
where we have introduced the new time variable T = T / Q measured in units of the relaxation time of the exponential. In addition, terms of order N-l and Q-l have been neglected in comparison with unity. The estimate for the momentum autocorrelation function po("(T) is useful provided that the error bound on the right-hand side of eq 19 is small compared to the exponential on the left-hand side. We will now examine the nature of the error estimate in (19) in more detail, First we consider the special case N = a, and then the general case of finite N . N = m and Q >> 1. In the limit N = a,it is readily verified that when T = a In Q, the ratio of the error
(R
Z Q - @ ( T cIn Y Q
+ '/z)
(22)
and the ratio a approaches zero in the limit Q -+ Q), N = For example, when a = 1, Q = lo4, and p = 1, the ratio a 2.94 X and T 2 9.21. In this example the correction to the exponential is less of the value of the exponential after nine than 3 X relaxation times. It is clear that, as Q increases and the T interval in which the exponential with N = -a In Q is a good approximation to P ~ ( ~ ) (increases T) and the error in the approximation approaches zero at least as fast at Q-("+@)In Q, where a > 0 and p a < 5/2. It is in the above sense that the momentum autocorrelation function of a heavy particle in a finite crystal is a simple exponential.
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Rubin j Momentum Autocorrelation Function of Heavy Particle in Finite Crystal
