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Lifshitz theory-— exact or approximate statistical mechanics?
Volume 48A, number 4
LIFSHITZ THEORY
PHYSICS LETTERS
—
1 July 1974
EXACT OR APPROXIMATE STATISTICAL MECHANICS? B. DAVIES
Department ofApplied Mathematics, School of General Studies, Australian National University, Australia Received 31 May 1974 We show that the Ljfshitz and related theories of intermolecular interactions are equivalent to perturbation theory, and therefore subject to the usual restrictions of the latter.
There has been a great deal of interest in recent years in the Lifshitz theory [1 —31 of van der Waals forces in condensed media, and its connection with dispersion relations [4—7],zero point energy [8—il] and statistical mechanics in general. Moreover, these questions are of considerable importance since the use of the Lifshitz theory in various applications is becom. ing more common [12], sometimes with very work good is agreement with experiment. Lifshitz’ original essentially a clever application of perturbation theory constructed as follows: the possible electromagnetic states of the system are assumed to be small perturbations from some specified equilibrium state; the latter being taken as the ‘average’ state. For a system composed of dielectrics, this average state is one of zero electric and magnetic fields. Now the system is constantly fluctuating between the available states, and the fluctuation-dissipation theorem [13] is used to obtain the statistical density of states and hence to calculate the interaction energy. In two recent notes [4—5], we have shown that there is a complete equivalence between this approach, normal mode methods, and the use of the statistical Green’s function method given in [31, p. 264. There has also been recent work by other authors in which a general formalism involving Green’s functions is developed from the basic principles of quantum statistical mechanics [14, 15], without making intermediate assumptions of the type made by Lifshitz. It is the purpose of this paper to discuss the connection between these latter results and Lifshitz theory; in particular we show that the two approaches are not equivalent, a conclusion which sheds light on the limitations of Lifshitz theory. The fundamental problem is to compute the inter-
free energy of the uncoupled system described by H0. By introducing a fictitious Hamiltonian H(X) = H0 ÷XFI~~(O ~ X~1), and using standard results of quantum statistical mechanics, Graig [14] and Mitchell and Richmond [15] have shown that this interaction energy may be written (we use the notation of [15]) 1 2~Tr[G(A)] (1) F.mt = —kT ~J ~x where the sum is over the frequencies wn = 27TiflkT/h, the prime on the summation sign means that the n = 0 term has weight one-half, Tr is the trace operation and G(X) is the retarded linear response operator which relates a small applied charge current system j~to the induced charges and currents iind via j~(r, w)
298
=
—JGMV(r,
r’, w)j~t(r’ w)dr’.
(2)
We emphasise the fact that (1) is derived without making assumptions additional to those inherent in the foundations of quantum statistucal mechanics. The purpose of this note is to compare this formalism with a similar result given in [3], p. 264, which after summation of the infinite series reads [5] F~,= kT ~
‘
Tr[lnD(1)
—
lnD(0)].
(3)
n0
Here D(X) is a different retarded linear response operator relating an applied charge current vector/~tto the induced potential change A,~via A, 2(r, w)
action free energy F~tof a system whose Hamiltonian has the splitting H = H0 + ~ relative to the known
r
~‘
=
—
~
r’, w)/ext(r’ w)dr’.
(4)
Now we may write (3) as an integral over X by consider-
Volume 48A, number 4
PHYSICS LETTERS
ing the identity lnD = fD—1(A)D’ (A)dx, where the prime denotes differentiation with respect to A, and using Dyson’s equation, which relates D (0), D (A) and the polarization operator 11(A) by D(A) = D(O) + D(0) 11(A)D(A), to replace D~(A) by D1 (0) —11(A). After integration by parts, this gives as an alternative for (3) F.m
=
E
1
f
—kT n”O ~ dA Tr [H’(A)D(A)J.
(5)
This is very similar to (1) since G (A) = 11 (A)D (A); furthermore we see that for equality it is sufficient to require that AH’(A) = 11(A), that is, that the polarization operator is a linear function of the coupling constant A(=e2). This imposes a fundamental restriction on the validity of Lifshitz theory, or any other results based on the use of (3), in particular all calculations using dispersion relations are only valid to within this approximation of the linearity of 11 as a function of e2. It is interesting to seek the reason for the nonequivalence of(1) and (3) within the framework of statistical Green~sfunctions. A full discussion of the terms (diagrams) which are included in the polarization operator H is given in [3] p. 261, where it is stated that no diagrams involving the non-linear proces of photon-photon scattering are to be included as an internal part of a contributing diagram. In fact, the only diagram which is excluded from this restriction is the simple bubble diagram corresponding to the bare interaction of a photon with an electron, the latter then returning to its original state without undergoing any further intermediate interactions. For if other diagrams are considered, they must have closed photon lines and therefore arise from closing a photon-photon scatfeting diagram, contrary to the stated restriction. Consequently, as is clear from the discussion on page 263 of [3], the expression which is derived for the interactions energy is simply the random phase approximation (RPA) within which 11 contains two electronphoton vertices and is therefore proportional to e2. Results which are obtained from (3) are not, however, equivalent to a simple use of the RPA, since in practicab applications (3) leads to a functional relation giving ~ in terms of the dielectric properties of the various media, and these dielectric properties are usually taken from experimental data rather than a model such as
1 July 1974
RPA.In any case, comparison of(1) and(3) shows that the actual limit of the validity of the latter is that the difference between the interacting and non-interacting states of the system being considered must be sufficiently small for the linearity assumption to hold. As stated above, (3) is essentially the result ofperturbation theory carried out with Green’s functions, and as such is sub-
-
ject to the usual limitations of perturbation results. Webased must on alsozero-point consider the connection with derivations energy considerations, particularly the paper of Renne [11], where the zero-temperature retarded Lifshitz formula is recovered from the Drude—Lorentz model. Since this model simply views the system as a set of coupled harmonic oscillators, the ground state energy must be the sum of the zero point energies of each mode; however, the difficulty encountered in Renne’s work is the diagonalisation of the coupled Hamiltonian. This is done by using the unperturbed wave functions for the photon field in order to remove them from the problem; hence the ensuing formulae are perturbation results, and the equivalence with the Lifshitz result is another indication of the approximate nature of the latter. In conclusion, we would stress the fact that our results do not detract from the usefulness of Lifshitz theory where a calculation based on perturbation theory is adequate. It is important to note in this context that the unperturbed state consists of non-interacting condensed bodies, so that there is no implication that we are restricted to dilute systems; in fact the interaction of interest may be exceedingly small compared with the internal interactions which hold the bodies together. Such situations presumably include the interactions of macroscopic bodies over distances which are considerably larger than the interactomic spacing, and are common in various branches of colbid science and biophysics. On the other hand, the difficulties raised in this note are not related to the practical problem of the adequacy of the physical model used for the linear response operators; they are fundamental, reflecting the intractability of the manybody problem to extract analysis. The beauty of the Lifshitz theory is that it allows an important calculation to be performed using observed data for the electromagnetic properties of the system, i.e., the observed values of D(1). In situations where there is a significant deviation from linearity of 11(X), this convenient simplification is lost. 299
Volume 48A, number 4
PHYSICS LETTERS
I would like to thank Professor B.W. Ninham and Dr. D.J. Mitchell for stimulating discussions, also Drs. D.J. Mitchell and P. Richmond for making available details of their work prior to publication.
1 July 1974
B.W. Ninhani and P. Richmond, J. Phys. C. (Solid State Phys.) 4 (1971) 1988. [81 D. Langbein, J. Phys. A: Gen. Phys. 4(1971)471; J. Pliys. Chem. Solids 32 (1971) 133. [9] M.J. Renne and B.R.A. Nijboer, Chem. Phys. Lett. 1 (1967) 317; B.R.A. Nijboer and M.J. Renne, Chem. Phys. Lett. 2
References
(1968) 35. Physica 53 (1971) 193. [10] M.J. Renne,
[1] E.M. Lifshitz, Soviet Phys. (JETP) 2(1956) 73. [2] I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Adv. Phys. 10 (1961) 165. [31 A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinskii, methods of quantum field theory in statistical mechanics (Prentice-Hall, 1963). [4] B. Davies, Chem. Phys. Lett. 16 (1972) 388. [51 B. Davies, Phys. Lett. 37A(1971) 391. 161 N.G. Van Kampen, B.R.A. Nijboer and K. Schram, Phys. Lett. 26A (1968) 307. [71 B.W. Ninham, V.A. Parsegian and G.H. Weiss, J. Stat. Phys. 2 (1970) 323;
[11] MJ. Renne, Physica56 (1971) 125. [12] B.W. Ninham and V.A. Parsegian, Biophys. J. 10 (1970) 646; V.A. Parsegian and B.W. Ninham, Biophys. J. 10 (1970) 664; P. Richmond and B.W. Ninham, J. Low Temp. Phys 5 (1971) 177; E.S. Sabisky and C.H. Anderson, Phys. Rev. A7 (1973) 790. [13] L.D. Landau and E.M. Lifshitz, Statistical physics, (Pergamon 1959), P. 400 ff. [14] R.A. Craig, Phys. Rev. B6 (1972) 1134. [151 D.J. Mitchell and P. Richmond, to be published.
300
LIFSHITZ THEORY
PHYSICS LETTERS
—
1 July 1974
EXACT OR APPROXIMATE STATISTICAL MECHANICS? B. DAVIES
Department ofApplied Mathematics, School of General Studies, Australian National University, Australia Received 31 May 1974 We show that the Ljfshitz and related theories of intermolecular interactions are equivalent to perturbation theory, and therefore subject to the usual restrictions of the latter.
There has been a great deal of interest in recent years in the Lifshitz theory [1 —31 of van der Waals forces in condensed media, and its connection with dispersion relations [4—7],zero point energy [8—il] and statistical mechanics in general. Moreover, these questions are of considerable importance since the use of the Lifshitz theory in various applications is becom. ing more common [12], sometimes with very work good is agreement with experiment. Lifshitz’ original essentially a clever application of perturbation theory constructed as follows: the possible electromagnetic states of the system are assumed to be small perturbations from some specified equilibrium state; the latter being taken as the ‘average’ state. For a system composed of dielectrics, this average state is one of zero electric and magnetic fields. Now the system is constantly fluctuating between the available states, and the fluctuation-dissipation theorem [13] is used to obtain the statistical density of states and hence to calculate the interaction energy. In two recent notes [4—5], we have shown that there is a complete equivalence between this approach, normal mode methods, and the use of the statistical Green’s function method given in [31, p. 264. There has also been recent work by other authors in which a general formalism involving Green’s functions is developed from the basic principles of quantum statistical mechanics [14, 15], without making intermediate assumptions of the type made by Lifshitz. It is the purpose of this paper to discuss the connection between these latter results and Lifshitz theory; in particular we show that the two approaches are not equivalent, a conclusion which sheds light on the limitations of Lifshitz theory. The fundamental problem is to compute the inter-
free energy of the uncoupled system described by H0. By introducing a fictitious Hamiltonian H(X) = H0 ÷XFI~~(O ~ X~1), and using standard results of quantum statistical mechanics, Graig [14] and Mitchell and Richmond [15] have shown that this interaction energy may be written (we use the notation of [15]) 1 2~Tr[G(A)] (1) F.mt = —kT ~J ~x where the sum is over the frequencies wn = 27TiflkT/h, the prime on the summation sign means that the n = 0 term has weight one-half, Tr is the trace operation and G(X) is the retarded linear response operator which relates a small applied charge current system j~to the induced charges and currents iind via j~(r, w)
298
=
—JGMV(r,
r’, w)j~t(r’ w)dr’.
(2)
We emphasise the fact that (1) is derived without making assumptions additional to those inherent in the foundations of quantum statistucal mechanics. The purpose of this note is to compare this formalism with a similar result given in [3], p. 264, which after summation of the infinite series reads [5] F~,= kT ~
‘
Tr[lnD(1)
—
lnD(0)].
(3)
n0
Here D(X) is a different retarded linear response operator relating an applied charge current vector/~tto the induced potential change A,~via A, 2(r, w)
action free energy F~tof a system whose Hamiltonian has the splitting H = H0 + ~ relative to the known
r
~‘
=
—
~
r’, w)/ext(r’ w)dr’.
(4)
Now we may write (3) as an integral over X by consider-
Volume 48A, number 4
PHYSICS LETTERS
ing the identity lnD = fD—1(A)D’ (A)dx, where the prime denotes differentiation with respect to A, and using Dyson’s equation, which relates D (0), D (A) and the polarization operator 11(A) by D(A) = D(O) + D(0) 11(A)D(A), to replace D~(A) by D1 (0) —11(A). After integration by parts, this gives as an alternative for (3) F.m
=
E
1
f
—kT n”O ~ dA Tr [H’(A)D(A)J.
(5)
This is very similar to (1) since G (A) = 11 (A)D (A); furthermore we see that for equality it is sufficient to require that AH’(A) = 11(A), that is, that the polarization operator is a linear function of the coupling constant A(=e2). This imposes a fundamental restriction on the validity of Lifshitz theory, or any other results based on the use of (3), in particular all calculations using dispersion relations are only valid to within this approximation of the linearity of 11 as a function of e2. It is interesting to seek the reason for the nonequivalence of(1) and (3) within the framework of statistical Green~sfunctions. A full discussion of the terms (diagrams) which are included in the polarization operator H is given in [3] p. 261, where it is stated that no diagrams involving the non-linear proces of photon-photon scattering are to be included as an internal part of a contributing diagram. In fact, the only diagram which is excluded from this restriction is the simple bubble diagram corresponding to the bare interaction of a photon with an electron, the latter then returning to its original state without undergoing any further intermediate interactions. For if other diagrams are considered, they must have closed photon lines and therefore arise from closing a photon-photon scatfeting diagram, contrary to the stated restriction. Consequently, as is clear from the discussion on page 263 of [3], the expression which is derived for the interactions energy is simply the random phase approximation (RPA) within which 11 contains two electronphoton vertices and is therefore proportional to e2. Results which are obtained from (3) are not, however, equivalent to a simple use of the RPA, since in practicab applications (3) leads to a functional relation giving ~ in terms of the dielectric properties of the various media, and these dielectric properties are usually taken from experimental data rather than a model such as
1 July 1974
RPA.In any case, comparison of(1) and(3) shows that the actual limit of the validity of the latter is that the difference between the interacting and non-interacting states of the system being considered must be sufficiently small for the linearity assumption to hold. As stated above, (3) is essentially the result ofperturbation theory carried out with Green’s functions, and as such is sub-
-
ject to the usual limitations of perturbation results. Webased must on alsozero-point consider the connection with derivations energy considerations, particularly the paper of Renne [11], where the zero-temperature retarded Lifshitz formula is recovered from the Drude—Lorentz model. Since this model simply views the system as a set of coupled harmonic oscillators, the ground state energy must be the sum of the zero point energies of each mode; however, the difficulty encountered in Renne’s work is the diagonalisation of the coupled Hamiltonian. This is done by using the unperturbed wave functions for the photon field in order to remove them from the problem; hence the ensuing formulae are perturbation results, and the equivalence with the Lifshitz result is another indication of the approximate nature of the latter. In conclusion, we would stress the fact that our results do not detract from the usefulness of Lifshitz theory where a calculation based on perturbation theory is adequate. It is important to note in this context that the unperturbed state consists of non-interacting condensed bodies, so that there is no implication that we are restricted to dilute systems; in fact the interaction of interest may be exceedingly small compared with the internal interactions which hold the bodies together. Such situations presumably include the interactions of macroscopic bodies over distances which are considerably larger than the interactomic spacing, and are common in various branches of colbid science and biophysics. On the other hand, the difficulties raised in this note are not related to the practical problem of the adequacy of the physical model used for the linear response operators; they are fundamental, reflecting the intractability of the manybody problem to extract analysis. The beauty of the Lifshitz theory is that it allows an important calculation to be performed using observed data for the electromagnetic properties of the system, i.e., the observed values of D(1). In situations where there is a significant deviation from linearity of 11(X), this convenient simplification is lost. 299
Volume 48A, number 4
PHYSICS LETTERS
I would like to thank Professor B.W. Ninham and Dr. D.J. Mitchell for stimulating discussions, also Drs. D.J. Mitchell and P. Richmond for making available details of their work prior to publication.
1 July 1974
B.W. Ninhani and P. Richmond, J. Phys. C. (Solid State Phys.) 4 (1971) 1988. [81 D. Langbein, J. Phys. A: Gen. Phys. 4(1971)471; J. Pliys. Chem. Solids 32 (1971) 133. [9] M.J. Renne and B.R.A. Nijboer, Chem. Phys. Lett. 1 (1967) 317; B.R.A. Nijboer and M.J. Renne, Chem. Phys. Lett. 2
References
(1968) 35. Physica 53 (1971) 193. [10] M.J. Renne,
[1] E.M. Lifshitz, Soviet Phys. (JETP) 2(1956) 73. [2] I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Adv. Phys. 10 (1961) 165. [31 A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinskii, methods of quantum field theory in statistical mechanics (Prentice-Hall, 1963). [4] B. Davies, Chem. Phys. Lett. 16 (1972) 388. [51 B. Davies, Phys. Lett. 37A(1971) 391. 161 N.G. Van Kampen, B.R.A. Nijboer and K. Schram, Phys. Lett. 26A (1968) 307. [71 B.W. Ninham, V.A. Parsegian and G.H. Weiss, J. Stat. Phys. 2 (1970) 323;
[11] MJ. Renne, Physica56 (1971) 125. [12] B.W. Ninham and V.A. Parsegian, Biophys. J. 10 (1970) 646; V.A. Parsegian and B.W. Ninham, Biophys. J. 10 (1970) 664; P. Richmond and B.W. Ninham, J. Low Temp. Phys 5 (1971) 177; E.S. Sabisky and C.H. Anderson, Phys. Rev. A7 (1973) 790. [13] L.D. Landau and E.M. Lifshitz, Statistical physics, (Pergamon 1959), P. 400 ff. [14] R.A. Craig, Phys. Rev. B6 (1972) 1134. [151 D.J. Mitchell and P. Richmond, to be published.
300