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On local thermal equilibrium given by an energy eigenstate in isolated quantum mechanical system
6 August 2001
Physics Letters A 286 (2001) 314–320 www.elsevier.com/locate/pla
On local thermal equilibrium given by an energy eigenstate in isolated quantum mechanical system Nobuyuki Arikawa ∗ , Tetsuji Tokihiro Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan Received 28 February 2001; received in revised form 4 June 2001; accepted 5 June 2001 Communicated by C.R. Doering
Abstract It is proved that almost all energy eigenstates of a certain isolated system have the following property: quantum expectation values of all local observables are equal to these statistical expectation values. Temperature is uniquely determined by energy density. 2001 Elsevier Science B.V. All rights reserved. PACS: 05.30.-d; 03.65.-w; 02.50.Cw Keywords: Boltzmann’s principle; Assumption of equal a priori probabilities; Quantum expectation value
An expectation value of an observable in a subsystem of an isolated system is supposed to be equal to its statistical expectation value, i.e., the expectation value given by the Boltzmann distribution of the subsystem, when the subsystem is in thermal equilibrium [1,2]. This hypothesis is usually called the Boltzmann’s principle. The most popular arguments to justify the Boltzmann’s principle in classical mechanics is as follows [3]. The ergodic property is extracted from the phase trajectory of the isolated system, and the assumption of equal a priori probabilities of the isolated system is justified by the ergodic property, and the Boltzmann distribution of the subsystem is derived from the assumption of equal a priori probabilities. The Boltzmann’s principle, however, is not justified in quantum mechanics in the same manner. Since the phase trajectory is not defined in quantum mechanics, we have to justify the assumption of equal a priori probabilities of the isolated system in a different manner in order to derive the Boltzmann distribution of the subsystem. In quantum mechanics, the assumption of equal a priori probabilities was justified only under the following conditions [4]. First, the subsystem and the rest of the isolated system are independent, and second, the observables and Hamiltonian of the subsystem are compatible. For systems and observables which do not satisfy these conditions, it is desirable to justify the Boltzmann’s principle without the assumption of equal a priori probabilities of the isolated system. There are several numerical results for this problem [5,6]. We present a mathematically rigorous result corresponding to some of these numerical results [7]. The steps to justify the Boltzmann’s principle in this Letter are as follows.
* Corresponding author.
E-mail address: [email protected] (N. Arikawa). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 0 7 - 8
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
315
Fig. 1. Two similar systems, N system and M system. Both of the two systems contain the area L, in which all the local observables are defined.
• Consider all the local observables defined in an area L. Consider two similar systems, N system and M system. Both of the two systems contain the area L and the local observables. The size of the N system, M system and area L is N , M, and L, respectively, where N M L (Fig. 1). • Choose an energy eigenstate of the N system which satisfies specific conditions (which are assumed in Proposition 2). Estimate the difference between quantum expectation value of each observable for the energy eigenstate of the N system and statistical expectation value for the M system (Proposition 2). • Prove that almost all energy eigenstates of the N system satisfy the specific conditions (Theorem 1). • Time-independent term of quantum expectation value of any observable for a state of N system is equal to superposition of quantum expectation values for energy eigenstates of the N system. Prove that, for a generic state of the N system, the superposition of quantum expectation values coincides with the statistical expectation value for the M system in the thermodynamic limit N → ∞, M → ∞ (Corollary 1). The N system is a one-dimensional lattice composed of N particles, where N is supposed to be odd for simplicity. The N system is described by Hamiltonian pj 2 1 1 2 2 + mω xj + g|j −j | (xj − xj )2 , H (N) = 2m 2 2 j ∈A j ∈A j ∈A N −1 N −1 j A := j ∈ Z − . 2 2 Here xj (pj ) are displacement (momentum) operators of the j th particle, m is a mass of each particle, ω is a frequency of harmonic potential, and g|j −j | are coupling constants of energy transfer between the j th particle and the j th particle. The periodic boundary condition is adopted. It is assumed that the energy transfer is local, i.e., there exists a length lg ( N) such that g|j −j | = 0 for |j − j | > lg . This Hamiltonian is rewritten to that of noninteracting harmonic oscillators 1 2π † Z −π < q < π . H (N) = hω , I := q ∈ ¯ q aq aq + 2 N q∈I
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N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
Here frequencies ωq and operators aq are given as 1/2 q 2 1 2 g|j | 2 sin j , ωq = ω + m 2 j ∈A mωq 1 i √ xj + mω pj cos qj , for q > 0, j ∈A h ¯ 0 N mω0 √1 aq = xj + mωi 0 pj , for q = 0, j ∈A 2 h ¯ N mωq √1 xj + i pj sin qj , for q < 0, j ∈A h¯ mω0 N
(. . .)†
denotes the Hermitian conjugate operator of (. . .). An eigenstate of the Hamiltonian H (N) is denoted and by |{nq }, which satisfies aq† aq |{nq } = nq |{nq } (∀q ∈ I ). Hence the eigenenergy associated to the eigenstate |{nq } is given by 1 E{nq } = h¯ ωq nq + . 2 q∈I
The M system is described by Hamiltonian 2π 1 † ˜ h¯ ωp ap ap + , I := p ∈ Z −π < p < π , H (M) = 2 M
(1)
p∈I˜
where M is odd and satisfies lg M N . In the thermodynamic limit N → ∞, M → ∞, this Hamiltonian becomes that of the N system. This Hamiltonian is also equivalent to that of the subsystem of the N system if we neglect the effect of the boundary, which is negligible in the thermodynamic limit N → ∞, M → ∞. As the area L, we introduce a set of local sites on which the observables are defined: L−1 L−1 j . AL := j ∈ Z − 2 2 Here L is a positive odd integer and L M. The observables are expressed as linear combinations of the products of coordinates and momentums at these local sites: nj n j nj n j sj xj exp tj pj , xj pj = ∂sj ∂tj exp j ∈AL
j ∈AL
j ∈AL
j ∈AL
sj ,tj =0 (∀j ∈AL )
where {nj }, {n j } are positive integers. To avoid treating all the observables, we introduce a generating function of these observables h¯ sj xj exp tj pj = exp ΩL := exp sj xj + tj pj + i sj tj 2 j ∈AL j ∈AL j ∈AL h¯ h¯ ∗ † ∗ † Dq aq + Dq aq + i Cp ap + Cp ap + i sj tj = exp sj tj , = exp 2 2 q∈I j ∈AL j ∈AL p∈I˜
ωq ωq ω0 ω0 ∗ √1 + + − ζ j N j ∈AL 2ωq 2ω0 2ωq 2ω0 ζj cos qj , for q > 0, 1 for q = 0, Dq := √N j ∈AL ζj ,
ωq ωq ω0 ω0 ∗ √1 for q < 0, j ∈AL 2ωq + 2ω0 ζj + 2ωq − 2ω0 ζj sin qj , N
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
ωp ω0 + √1 j ∈A 2ωp 2ω0 L M 1 Cp := √M j ∈AL ζj ,
ωp ω0 √1 + j ∈AL 2ωp 2ω0 M h¯ mh¯ ω0 tj . sj − i ζj := 2mω0 2
317
ωp ω0 ∗ ζj + − 2ωp 2ω0 ζj cos pj , for p > 0,
ωp ω0 ∗ − ζj + 2ωp 2ω0 ζj sin pj ,
for p = 0, for p < 0,
It is clear that if a statement holds for the generating function for arbitrary {tj }, {sj }, then, it also holds for all the observables. First we evaluate the statistical expectation value of the generating function for the M system. The statistical expectation value is easily obtained as h¯ Tr[ΩL exp(−βH (M))] 1 = exp |Cp |2 n¯ βp + sj tj , ΩL := (2) +i Tr[exp(−βH (M))] 2 2 j ∈AL
p∈I˜
where β is inverse temperature and 1 . eβ h¯ ωp − 1 Now we take an energy eigenstate √ of the N system and consider its quantum expectation value of the generating function. Noticing |Dq | = O(1/ N ), the quantum expectation value is evaluated as Dq aq +D ∗ aq† h¯ q nq exp i nq e sj tj {nq } ΩL {nq } = 2 q∈I j ∈AL 2 h¯ |Dq |2 (nq +1/2) 4 1 + O nq |Dq | exp i sj tj e = 2 n¯ βp :=
j ∈AL
q∈I
2 nq h¯ 1 2 +i = exp 1+O . |Dq | nq + sj tj 2 2 N2 q∈I
j ∈AL
(3)
q∈I
√ The product q∈I {1 + O(n2q /N 2 )} in (3) is nearly equal to one, 1 + o(1) (N → ∞), when nq = o( N ) (∀q ∈ I ). We divide the set I into subsets Ip as ! π π 2π Zp −
Since the coefficients Cp and Dq have a relation M Cp , Dq |q=p = N the coefficients Dq are estimated as lg L M Cp 1 + O +O (∀q ∈ Ip ). Dq = N M M Hence we find that the terms in (3) which contain the coefficient Dq are described by the coefficient Cp , lg 1 1 L 2 2 |Dq | nq + |Cp | n¯ p + = 1+O +O , 2 2 M M q∈I
p∈I˜
(4)
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N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
where n¯ p is an average of quantum numbers among q ∈ Ip : 1 n¯ p = nq . N/M q∈Ip
From (2), (3) and (4), the difference between the statistical expectation value of the generating function for the M β system and quantum expectation value for an energy eigenstate of the N system which satisfies n¯ p = n¯ p (∀p ∈ I˜) is estimated: √ β Proposition 1. If nq = o( N ) (∀q ∈ I ) and n¯ p = n¯ p (∀p ∈ I˜ ), lg L {nq } ΩL {nq } = ΩL 1 + O +O + o(1) (N → ∞) . M M ¯ We consider a set of energy eigenstates We introduce an energy density of the N system E¯ and its fluctuation δ E. β whose averages of quantum numbers n¯ p are nearly equal to n¯ p , ¯ ¯ δ E, ¯ N : M) := {nq } n¯ p − n¯ βp (M/N)1/4 + C0 δ E ∀p ∈ I˜ , S ∗ (E, E¯ β
where C0 (< +∞) is a positive constant which depends neither M nor N and the inverse temperature β in n¯ p is a function of M, E¯ and is determined by # " H (M) ¯ = E. (5) M The difference between the statistical expectation value of the generating function and quantum expectation value for an energy eigenstate in the set S ∗ is estimated: √ Proposition 2. If nq = o( N ) (∀q ∈ I ) and {nq } ∈ S ∗ , then ¯ 1/4 lg δE L M {nq } ΩL {nq } = ΩL 1 + O +O +O +O + o(1) (N → ∞) . M M N 1/4 E¯ In fact, the set S ∗ is nearly equal to a set of energy eigenstates whose energy density is equal to E¯ within the ¯ small fluctuation δ E: ¯ δ E, ¯ N) = {nq } E{nq } − N E¯ Nδ E¯ . S(E, Theorem 1. |S \ S ∗ | = o(1) |S|
lg E¯ N →∞ . → 0 + o(1) M Mδ E¯
Brief sketch of the proof. To prove the theorem, we have to elaborate on lengthy computation, so we show a brief sketch of the proof here. We introduce another two sets of energy eigenstates ¯ δ E, ¯ N : M, η) := {nq } E˜ {n¯ p } − M E¯ (1 ± η)Mδ E¯ , S˜± (E, where E˜ {n¯ p } :=
p∈I˜
1 h¯ ωp n¯ p + , 2
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
319
and η is a small parameter which satisfies lg E¯ η=O . Mδ E¯ From the condition of the parameter η, we find that the set S satisfies S˜− ⊂ S ⊂ S˜+ . Therefore |S ∩ S ∗ | |S˜− ∩ S ∗ | |S˜− ∩ S ∗ | = 1 + o(1) (η → 0) . > |S| |S˜+ | |S˜− | From explicit evaluation of the number of elements in S˜− , we can show that S˜− = S˜− ∩ S ∗ 1 + o(1) (N/M → ∞) , which completes the proof. ✷ We take an arbitrary linear combination of the energy eigenstates in the set S, Φ(0) = c{nq } {nq } , {nq }∈S
and consider its quantum expectation value of the generating function. The quantum expectation value at time t is given as |c{nq } |2 {nq }ΩL {nq } Φ(t)ΩL Φ(t) = {nq }∈S
+
{nq }∈S
{n q }∈S {n q } ={nq }
E {nq } − E{n q } ∗ t {nq }ΩL {n q } . c{n c exp i q } {nq } h¯
(6)
The first term of the right-hand side of (6) is the √superposition of quantum expectation values of energy eigenstates. Since a generic states satisfies that nq = o( N ) (∀q ∈ I ), from Proposition 2 and Theorem 1, we state the following corollary: Corollary 1. For a generic state of the N system, the time-independent term of the quantum expectation values of all observables in the area L are equal to the statistical expectation values for the M system in the thermodynamic ¯ E¯ → 0. limit N → ∞, M → ∞ and δ E/ Numerical results [5,6] tell us that the time-dependent term of the quantum expectation value (6) decay after proper time [8]. From these results and Corollary 1, we expect that the quantum expectation value (6) approach to the statistical expectation value after proper time. It is interesting future problem to extend this method to various systems, so that we obtain local correlation functions in those systems at finite temperature.
Acknowledgements The authors would like to thank Seiji Miyashita, Keiji Saito and Hal Tasaki for helpful comments. This work was supported in part by a Grant-in-Aid from the Japan Ministry of Education, Science and Culture.
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References [1] [2] [3] [4] [5] [6] [7] [8]
R.P. Feynman, Statistical Mechanics, Addison-Wesley, 1972. L.D. Landau, E.M. Lifshitz, Statistical Physics Part 1, 3rd ed., Pergamon Press, 1980. S. Tomonaga, Quantum Mechanics, Vol. I, North-Holland, 1962, Appendix I. A.Y. Khinchin, Mathematical Foundation of Quantum Statistics, Graylock Press, 1960. R.V. Jensen, R. Shankar, Phys. Rev. Lett. 54 (1985) 1879. K. Saito, S. Takesue, S. Miyashita, J. Phys. Soc. Jpn. 65 (1996) 1243. For another approach to this problem, see H. Tasaki, Phys. Rev. Lett. 80 (1998) 1373. These numerical results do not depend on whether the isolated system is integrable or not.
Physics Letters A 286 (2001) 314–320 www.elsevier.com/locate/pla
On local thermal equilibrium given by an energy eigenstate in isolated quantum mechanical system Nobuyuki Arikawa ∗ , Tetsuji Tokihiro Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan Received 28 February 2001; received in revised form 4 June 2001; accepted 5 June 2001 Communicated by C.R. Doering
Abstract It is proved that almost all energy eigenstates of a certain isolated system have the following property: quantum expectation values of all local observables are equal to these statistical expectation values. Temperature is uniquely determined by energy density. 2001 Elsevier Science B.V. All rights reserved. PACS: 05.30.-d; 03.65.-w; 02.50.Cw Keywords: Boltzmann’s principle; Assumption of equal a priori probabilities; Quantum expectation value
An expectation value of an observable in a subsystem of an isolated system is supposed to be equal to its statistical expectation value, i.e., the expectation value given by the Boltzmann distribution of the subsystem, when the subsystem is in thermal equilibrium [1,2]. This hypothesis is usually called the Boltzmann’s principle. The most popular arguments to justify the Boltzmann’s principle in classical mechanics is as follows [3]. The ergodic property is extracted from the phase trajectory of the isolated system, and the assumption of equal a priori probabilities of the isolated system is justified by the ergodic property, and the Boltzmann distribution of the subsystem is derived from the assumption of equal a priori probabilities. The Boltzmann’s principle, however, is not justified in quantum mechanics in the same manner. Since the phase trajectory is not defined in quantum mechanics, we have to justify the assumption of equal a priori probabilities of the isolated system in a different manner in order to derive the Boltzmann distribution of the subsystem. In quantum mechanics, the assumption of equal a priori probabilities was justified only under the following conditions [4]. First, the subsystem and the rest of the isolated system are independent, and second, the observables and Hamiltonian of the subsystem are compatible. For systems and observables which do not satisfy these conditions, it is desirable to justify the Boltzmann’s principle without the assumption of equal a priori probabilities of the isolated system. There are several numerical results for this problem [5,6]. We present a mathematically rigorous result corresponding to some of these numerical results [7]. The steps to justify the Boltzmann’s principle in this Letter are as follows.
* Corresponding author.
E-mail address: [email protected] (N. Arikawa). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 0 7 - 8
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
315
Fig. 1. Two similar systems, N system and M system. Both of the two systems contain the area L, in which all the local observables are defined.
• Consider all the local observables defined in an area L. Consider two similar systems, N system and M system. Both of the two systems contain the area L and the local observables. The size of the N system, M system and area L is N , M, and L, respectively, where N M L (Fig. 1). • Choose an energy eigenstate of the N system which satisfies specific conditions (which are assumed in Proposition 2). Estimate the difference between quantum expectation value of each observable for the energy eigenstate of the N system and statistical expectation value for the M system (Proposition 2). • Prove that almost all energy eigenstates of the N system satisfy the specific conditions (Theorem 1). • Time-independent term of quantum expectation value of any observable for a state of N system is equal to superposition of quantum expectation values for energy eigenstates of the N system. Prove that, for a generic state of the N system, the superposition of quantum expectation values coincides with the statistical expectation value for the M system in the thermodynamic limit N → ∞, M → ∞ (Corollary 1). The N system is a one-dimensional lattice composed of N particles, where N is supposed to be odd for simplicity. The N system is described by Hamiltonian pj 2 1 1 2 2 + mω xj + g|j −j | (xj − xj )2 , H (N) = 2m 2 2 j ∈A j ∈A j ∈A N −1 N −1 j A := j ∈ Z − . 2 2 Here xj (pj ) are displacement (momentum) operators of the j th particle, m is a mass of each particle, ω is a frequency of harmonic potential, and g|j −j | are coupling constants of energy transfer between the j th particle and the j th particle. The periodic boundary condition is adopted. It is assumed that the energy transfer is local, i.e., there exists a length lg ( N) such that g|j −j | = 0 for |j − j | > lg . This Hamiltonian is rewritten to that of noninteracting harmonic oscillators 1 2π † Z −π < q < π . H (N) = hω , I := q ∈ ¯ q aq aq + 2 N q∈I
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N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
Here frequencies ωq and operators aq are given as 1/2 q 2 1 2 g|j | 2 sin j , ωq = ω + m 2 j ∈A mωq 1 i √ xj + mω pj cos qj , for q > 0, j ∈A h ¯ 0 N mω0 √1 aq = xj + mωi 0 pj , for q = 0, j ∈A 2 h ¯ N mωq √1 xj + i pj sin qj , for q < 0, j ∈A h¯ mω0 N
(. . .)†
denotes the Hermitian conjugate operator of (. . .). An eigenstate of the Hamiltonian H (N) is denoted and by |{nq }, which satisfies aq† aq |{nq } = nq |{nq } (∀q ∈ I ). Hence the eigenenergy associated to the eigenstate |{nq } is given by 1 E{nq } = h¯ ωq nq + . 2 q∈I
The M system is described by Hamiltonian 2π 1 † ˜ h¯ ωp ap ap + , I := p ∈ Z −π < p < π , H (M) = 2 M
(1)
p∈I˜
where M is odd and satisfies lg M N . In the thermodynamic limit N → ∞, M → ∞, this Hamiltonian becomes that of the N system. This Hamiltonian is also equivalent to that of the subsystem of the N system if we neglect the effect of the boundary, which is negligible in the thermodynamic limit N → ∞, M → ∞. As the area L, we introduce a set of local sites on which the observables are defined: L−1 L−1 j . AL := j ∈ Z − 2 2 Here L is a positive odd integer and L M. The observables are expressed as linear combinations of the products of coordinates and momentums at these local sites: nj n j nj n j sj xj exp tj pj , xj pj = ∂sj ∂tj exp j ∈AL
j ∈AL
j ∈AL
j ∈AL
sj ,tj =0 (∀j ∈AL )
where {nj }, {n j } are positive integers. To avoid treating all the observables, we introduce a generating function of these observables h¯ sj xj exp tj pj = exp ΩL := exp sj xj + tj pj + i sj tj 2 j ∈AL j ∈AL j ∈AL h¯ h¯ ∗ † ∗ † Dq aq + Dq aq + i Cp ap + Cp ap + i sj tj = exp sj tj , = exp 2 2 q∈I j ∈AL j ∈AL p∈I˜
ωq ωq ω0 ω0 ∗ √1 + + − ζ j N j ∈AL 2ωq 2ω0 2ωq 2ω0 ζj cos qj , for q > 0, 1 for q = 0, Dq := √N j ∈AL ζj ,
ωq ωq ω0 ω0 ∗ √1 for q < 0, j ∈AL 2ωq + 2ω0 ζj + 2ωq − 2ω0 ζj sin qj , N
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
ωp ω0 + √1 j ∈A 2ωp 2ω0 L M 1 Cp := √M j ∈AL ζj ,
ωp ω0 √1 + j ∈AL 2ωp 2ω0 M h¯ mh¯ ω0 tj . sj − i ζj := 2mω0 2
317
ωp ω0 ∗ ζj + − 2ωp 2ω0 ζj cos pj , for p > 0,
ωp ω0 ∗ − ζj + 2ωp 2ω0 ζj sin pj ,
for p = 0, for p < 0,
It is clear that if a statement holds for the generating function for arbitrary {tj }, {sj }, then, it also holds for all the observables. First we evaluate the statistical expectation value of the generating function for the M system. The statistical expectation value is easily obtained as h¯ Tr[ΩL exp(−βH (M))] 1 = exp |Cp |2 n¯ βp + sj tj , ΩL := (2) +i Tr[exp(−βH (M))] 2 2 j ∈AL
p∈I˜
where β is inverse temperature and 1 . eβ h¯ ωp − 1 Now we take an energy eigenstate √ of the N system and consider its quantum expectation value of the generating function. Noticing |Dq | = O(1/ N ), the quantum expectation value is evaluated as Dq aq +D ∗ aq† h¯ q nq exp i nq e sj tj {nq } ΩL {nq } = 2 q∈I j ∈AL 2 h¯ |Dq |2 (nq +1/2) 4 1 + O nq |Dq | exp i sj tj e = 2 n¯ βp :=
j ∈AL
q∈I
2 nq h¯ 1 2 +i = exp 1+O . |Dq | nq + sj tj 2 2 N2 q∈I
j ∈AL
(3)
q∈I
√ The product q∈I {1 + O(n2q /N 2 )} in (3) is nearly equal to one, 1 + o(1) (N → ∞), when nq = o( N ) (∀q ∈ I ). We divide the set I into subsets Ip as ! π π 2π Zp −
Since the coefficients Cp and Dq have a relation M Cp , Dq |q=p = N the coefficients Dq are estimated as lg L M Cp 1 + O +O (∀q ∈ Ip ). Dq = N M M Hence we find that the terms in (3) which contain the coefficient Dq are described by the coefficient Cp , lg 1 1 L 2 2 |Dq | nq + |Cp | n¯ p + = 1+O +O , 2 2 M M q∈I
p∈I˜
(4)
318
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
where n¯ p is an average of quantum numbers among q ∈ Ip : 1 n¯ p = nq . N/M q∈Ip
From (2), (3) and (4), the difference between the statistical expectation value of the generating function for the M β system and quantum expectation value for an energy eigenstate of the N system which satisfies n¯ p = n¯ p (∀p ∈ I˜) is estimated: √ β Proposition 1. If nq = o( N ) (∀q ∈ I ) and n¯ p = n¯ p (∀p ∈ I˜ ), lg L {nq } ΩL {nq } = ΩL 1 + O +O + o(1) (N → ∞) . M M ¯ We consider a set of energy eigenstates We introduce an energy density of the N system E¯ and its fluctuation δ E. β whose averages of quantum numbers n¯ p are nearly equal to n¯ p , ¯ ¯ δ E, ¯ N : M) := {nq } n¯ p − n¯ βp (M/N)1/4 + C0 δ E ∀p ∈ I˜ , S ∗ (E, E¯ β
where C0 (< +∞) is a positive constant which depends neither M nor N and the inverse temperature β in n¯ p is a function of M, E¯ and is determined by # " H (M) ¯ = E. (5) M The difference between the statistical expectation value of the generating function and quantum expectation value for an energy eigenstate in the set S ∗ is estimated: √ Proposition 2. If nq = o( N ) (∀q ∈ I ) and {nq } ∈ S ∗ , then ¯ 1/4 lg δE L M {nq } ΩL {nq } = ΩL 1 + O +O +O +O + o(1) (N → ∞) . M M N 1/4 E¯ In fact, the set S ∗ is nearly equal to a set of energy eigenstates whose energy density is equal to E¯ within the ¯ small fluctuation δ E: ¯ δ E, ¯ N) = {nq } E{nq } − N E¯ Nδ E¯ . S(E, Theorem 1. |S \ S ∗ | = o(1) |S|
lg E¯ N →∞ . → 0 + o(1) M Mδ E¯
Brief sketch of the proof. To prove the theorem, we have to elaborate on lengthy computation, so we show a brief sketch of the proof here. We introduce another two sets of energy eigenstates ¯ δ E, ¯ N : M, η) := {nq } E˜ {n¯ p } − M E¯ (1 ± η)Mδ E¯ , S˜± (E, where E˜ {n¯ p } :=
p∈I˜
1 h¯ ωp n¯ p + , 2
N. Arikawa, T. Tokihiro / Physics Letters A 286 (2001) 314–320
319
and η is a small parameter which satisfies lg E¯ η=O . Mδ E¯ From the condition of the parameter η, we find that the set S satisfies S˜− ⊂ S ⊂ S˜+ . Therefore |S ∩ S ∗ | |S˜− ∩ S ∗ | |S˜− ∩ S ∗ | = 1 + o(1) (η → 0) . > |S| |S˜+ | |S˜− | From explicit evaluation of the number of elements in S˜− , we can show that S˜− = S˜− ∩ S ∗ 1 + o(1) (N/M → ∞) , which completes the proof. ✷ We take an arbitrary linear combination of the energy eigenstates in the set S, Φ(0) = c{nq } {nq } , {nq }∈S
and consider its quantum expectation value of the generating function. The quantum expectation value at time t is given as |c{nq } |2 {nq }ΩL {nq } Φ(t)ΩL Φ(t) = {nq }∈S
+
{nq }∈S
{n q }∈S {n q } ={nq }
E {nq } − E{n q } ∗ t {nq }ΩL {n q } . c{n c exp i q } {nq } h¯
(6)
The first term of the right-hand side of (6) is the √superposition of quantum expectation values of energy eigenstates. Since a generic states satisfies that nq = o( N ) (∀q ∈ I ), from Proposition 2 and Theorem 1, we state the following corollary: Corollary 1. For a generic state of the N system, the time-independent term of the quantum expectation values of all observables in the area L are equal to the statistical expectation values for the M system in the thermodynamic ¯ E¯ → 0. limit N → ∞, M → ∞ and δ E/ Numerical results [5,6] tell us that the time-dependent term of the quantum expectation value (6) decay after proper time [8]. From these results and Corollary 1, we expect that the quantum expectation value (6) approach to the statistical expectation value after proper time. It is interesting future problem to extend this method to various systems, so that we obtain local correlation functions in those systems at finite temperature.
Acknowledgements The authors would like to thank Seiji Miyashita, Keiji Saito and Hal Tasaki for helpful comments. This work was supported in part by a Grant-in-Aid from the Japan Ministry of Education, Science and Culture.
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References [1] [2] [3] [4] [5] [6] [7] [8]
R.P. Feynman, Statistical Mechanics, Addison-Wesley, 1972. L.D. Landau, E.M. Lifshitz, Statistical Physics Part 1, 3rd ed., Pergamon Press, 1980. S. Tomonaga, Quantum Mechanics, Vol. I, North-Holland, 1962, Appendix I. A.Y. Khinchin, Mathematical Foundation of Quantum Statistics, Graylock Press, 1960. R.V. Jensen, R. Shankar, Phys. Rev. Lett. 54 (1985) 1879. K. Saito, S. Takesue, S. Miyashita, J. Phys. Soc. Jpn. 65 (1996) 1243. For another approach to this problem, see H. Tasaki, Phys. Rev. Lett. 80 (1998) 1373. These numerical results do not depend on whether the isolated system is integrable or not.