UUniversal relation for kinetic energy of quantum systems atequilibrium
Jerzy Łuczka
Institute of Physics, University of Silesia, 41-500 Chorzów, PolandInstitute of Physics, University of Augsburg, 86135 Augsburg, Germany
It is shown that the recently proposedquantum analogue of classical energyequipartition theorem for a free Brownianparticle and a dissipative harmonic oscilla-tor also holds true for all quantum systemscomposed of an arbitrary number of inter-acting particles, subjected to any confiningpotential and coupled to thermostat of ar-bitrary strength.
In classical statistical physics, the theorem onequipartition of kinetic energy is one of the mostuniversal relation [1, 2]. It states that for a sys-tem in thermodynamic equilibrium of tempera-ture T , the mean kinetic energy E k per one degreeof freedom is equal to E k = k B T / , where k B isthe Boltzmann constant [3, 4]. It does not dependon a number of particles in the system, the formof the potential force which acts on them, theform of interaction between particles and strengthof coupling between the system and thermostat.It depends only on the thermostat temperature T . On the contrary, for quantum systems, themean kinetic energy is not equally shared amongall degrees of freedom and the theorem fails. Thequestion arises whether one can formulate a sim-ilar and universal relation for the mean kineticenergy of quantum systems at a thermodynamicequilibrium state. Recently, in a series of papers[5, 6, 7], the authors have proposed quantum ana-logue of the energy equipartition theorem. For asystem of one degree of freedom this quantumcounterpart, which is called the energy partition theorem, has the appealing form: E k = Z ∞ E k ( ω ) P ( ω ) dω, (1) where E k ( ω ) = (cid:126) ω (cid:20) (cid:126) ω k B T (cid:21) (2) has the same form as the average kinetic energyof the harmonic oscillator with the frequency ω weakly coupled to thermostat [8] and P ( ω ) is aprobability density on a positive half-line of realnumbers meaning that P ( ω ) ≥ , (3) Z ∞ P ( ω ) dω = 1 . (4) The explicit form of P ( ω ) has been derived for twoexactly solved quantum systems: a free Brown-ian particle [5] and a dissipative harmonic oscilla-tor [6]. In these papers, thermostat is composedof quantum harmonic oscillators (à la Caldeira-Leggett [9]) and the above interpretation of E k ( ω ) is fully justified. Because P ( ω ) is a probabilitydensity, Eq. (1) can be rewritten in the form E k = hE k i , (5) where hE k i is a mean value of the function E k ( ξ ) of some random variable ξ distributed accordingto the probability density P . In the Caldeira-Leggett model, ξ can be interpreted as a ran-dom frequency of harmonic oscillators formingthe thermostat.Here, we want to prove a relation similar to(1) for a class of quantum systems for which theconcept of kinetic energy has sense (e.g spin sys-tems are outside of this class). More precisely, westudy a quantum system S coupled to thermo-stat (heat bath, environment) E . The compositesystem S + E is in a Gibbs equilibrium state oftemperature T defined by the density operator ρ = Z − e − H/k B T , Z = Tr h e − H/k B T i (6) and H = H S + H int + H E (7) is the Hamiltonian of the composite system S + E .Next, H S = X j p j M j + X j U S ( x j ) + X j,k V S ( x j , x k ) (8) a r X i v : . [ qu a n t - ph ] D ec s the Hamiltonian of the system S and H int = X j,n λ jn V ( x j , X n ) (9) is the Hamiltonian of interaction of the system S with the thermostat E . Finally, H E is theHamiltonian of thermostat E . Its explicit formis now not relevant. The set of parameters { λ jn } characterizes the coupling strength. The coordi-nate and momentum operators { x j , p j } refer tothe system S and the operators { X n } refer to thethermostat. All coordinate and momentum op-erators obey canonical equal-time commutationrelations. We assume that all components of theHamiltonian (7) fulfil required conditions to en-sure a well defined thermodynamic equilibriumstate of the composite system S + E . Theorem : The mean kinetic energy per onedegree of freedom of the system S can be pre-sented in a universal form as E ( j ) k = h p j M j i = Tr " p j M j ρ = Z ∞ E k ( ω ) P j ( ω ) dω, (10) where E k ( ω ) is given by Eq. (2) and P j ( ω ) isa probability density which obeys conditions (3)and (4).To prove the relation (10), we apply thefluctuation-dissipation relation of the Callen-Welton type [10, 11]. One can exploit the resultsderived e.g. in the Landau-Lifshitz book [12] [seeEq. (124.10)] or in the Zubarev book [13] [seeEq. (17.19g)] for the momentum operator p j ofthe system S . Without loss of generality we as-sume that the average momentum h p j i = 0 at theequilibrium state and then one obtains h p j i = (cid:126) π Z ∞ coth (cid:20) (cid:126) ω k B T (cid:21) χ jj ( ω ) dω (11) where the odd function χ jj ( ω ) is the imaginarypart of the generalized susceptibility, χ jj ( ω ) = χ jj ( ω ) + iχ jj ( ω ) . (12) In turn, the generalized susceptibility χ jj ( ω ) isthe Fourier transform χ jj ( ω ) = Z ∞−∞ e iωt G jj ( t ) dt (13) of the response function G jj ( t ) which in fact is theretarded thermodynamic Green function [13]: G jj ( t ) = i (cid:126) θ ( t ) h [ p j ( t ) , p j (0)] i , (14) where θ ( t ) is the Heaviside step function and p j ( t ) = exp( iHt/ (cid:126) ) p j (0) exp( − iHt/ (cid:126) ) (15) is the Heisenberg representation of the momen-tum p j (0) . The averaging in Eq. (14) is over theGibbs canonical statistical operator (6).Now, we compare Eqs. (10) and (11), and ob-tain the expression for P j ( ω ) in the form P j ( ω ) = 2 πM j χ jj ( ω ) ω . (16) The question is whether this function can be in-terpreted as a probability density. Its positivityfollows from the spectral representation of χ jj ( ω ) ,see the equation just above Eq. (124.9) in theLandau-Lifshitz book [12]. See also the text be-low Eq. (123.11) therein. The problem is to provethe normalization of (16): Z ∞ P j ( ω ) dω = 1 M j π Z ∞ χ jj ( ω ) ω dω. (17) To this aim, we can apply Eq. (123.19) in theLandau-Lifshitz book [12] which reads χ jj ( iω ) = 2 π Z ∞ uχ jj ( u ) ω + u du (18) (alternatively, one can apply the Kramers-Kronigdispersion relation). For ω = 0 it takes the form χ jj (0) = 2 π Z ∞ χ jj ( u ) u du. (19) On the other hand, from Eqs. (13) and (14) itfollows that χ jj (0) = Z ∞−∞ G jj ( t ) dt = i (cid:126) Z ∞ h [ p j ( t ) , p j (0)] i dt. (20) We notice that now the problem of normalizationof P j ( ω ) in Eq. (16) is converted to the problemwhether the equality χ jj (0) = M j (21) holds true for the Hamiltonian (7)-(9). This mayseem surprising at first glance since χ jj (0) doesnot depend on the form of the potential, interac-tion, temperature and parameters of the Hamil-tonian, but it depends only on mass M j .To prove the relation (21), we start from theHeisenberg equations of motion for coordinate op-erators of the system S , namely, dx j ( t ) dt = i (cid:126) [ H, x j ( t )] = p j ( t ) M j . (22) e insert it into Eq. (20) and obtain χ jj (0) = iM j (cid:126) lim (cid:15) → + Z ∞ e − (cid:15)t ddt h [ x j ( t ) , p j (0)] i dt = iM j (cid:126) e − (cid:15)t h [ x j ( t ) , p j (0)] i| ∞ + iM j (cid:126) lim (cid:15) → + (cid:15) Z ∞ e − (cid:15)t h [ x j ( t ) , p j (0)] i dt, (23) where we use a well-known limiting procedurewith the (cid:15) -term to ensure convergence of theintegral [14]. The integral in the last line is finiteand therefore the expression in the last line tendsto zero as (cid:15) → . In the middle line, for theupper limit t → ∞ the expression tends to zero.For the lower limit, h [ x j (0) , p j (0)] i = i (cid:126) . Thus itfinishes the proof of the relation (21). REMARKS :1. The formula (10) is a generalization of theenergy equipartition theorem. Indeed, in the hightemperature limit coth (cid:20) (cid:126) ω k B T (cid:21) ≈ k B T (cid:126) ω , E k ( ω ) ≈ k B T / and Eq. (10) reduces to its classical counterpart E ( j ) k = 12 k B T Z ∞ P j ( ω ) dω = 12 k B T (25) because of normalization of P j ( ω ) . We want tonotice that Callen and Welton in their ’histori-cal’ paper [10] missed the normalization: see Eq.(4.11) therein.2. It has to be stressed that the formula (10) isuniversal, however, the mean kinetic energy E ( j ) k depends not only on temperature of the system(as in the classical case) but also, via the proba-bility density P j ( ω ) , on a number of particles inthe system, the form of the potential which actson them, the form of interaction between parti-cles and strength of coupling between the systemand thermostat.3. If H is the Hamiltonian of the composite sys-tem S + E then all regimes, from weak to strongcoupling with thermostat, can be analyzed. How-ever, if H = H S (there is no explicit interactionwith thermostat) then it means that only theweak coupling limit can be considered becauseaveraging is over the Gibbs canonical density op-erator ρ S ∝ exp ( − H S /k B T ) valid for the weakcoupling limit. 4. There are no specific assumptions regardingthermostat E : It should be infinitely extendedand satisfying the Kubo-Martin-Schwinger condi-tions expressing periodicity of Green’s functionsin imaginary time [14, 15].5. The factor E k ( ω ) in Eq. (10) is the same asmean kinetic energy of a quantum harmonic os-cillator in the Gibbs state ρ O ∝ exp ( − H O /k B T ) ,where H O is the Hamiltonian of the harmonic os-cillator [8], E k ( ω ) = 12 m h p i = (cid:126) ω (cid:126) ω k B T . (26)
It depends on the frequency ω of the harmonicoscillator but not upon its mass m . However, inthe considered model (7)-(9), a harmonic oscilla-tor does not occur at all. It is a consequence ofthe above point 4 and the linear response theory[16].6. As an example, we demonstrate how theabove theory works for a free Brownian parti-cle coupled to thermostat which is a collection ofharmonic oscillators [5]. What we need is the ex-plicit form of the momentum operator p ( t ) whichhas been calculated e.g. in Ref. [5], see Eq. (7)therein. It reads p ( t ) = R ( t ) p (0) − Z t R ( t − u ) γ ( u ) du x (0)+ Z t R ( t − u ) η ( u ) du, (27) where R ( t ) and γ ( t ) are the response function andthe memory kernel of the generalized Langevinequation. The operator η ( t ) models quantumthermal noise and is expressed by thermostat op-erators which commute with the system opera-tors. In Eq. (14), only the second term in r.h.s.of Eq. (27) contributes to the commutator yield-ing the Green function G ( t ) = θ ( t ) Z t R ( t − u ) γ ( u ) du. (28) The susceptibility χ ( ω ) is a Fourier transform ofthe Green function G r ( t ) which is a convolution in(28) of two scalar functions R ( t ) and γ ( t ) . There-fore as a result we obtain χ ( ω ) = ˆ R L ( − iω )ˆ γ L ( − iω ) , (29) i.e., it is expressed by a product of two Laplacetransforms ˆ R L ( z ) and ˆ γ L ( z ) of the functions R ( t ) nd γ ( t ) , respectively. For the free Brownian par-ticle of mass M the Laplace transforms of R ( t ) reads [5] ˆ R L ( z ) = MM z + ˆ γ L ( z ) (30) and the generalized susceptibility takes the form χ ( ω ) = M ˆ γ L ( − iω ) − iωM + ˆ γ L ( − iω ) . (31) It is seen that for any form of the memory func-tion γ ( t ) the value of susceptibility at zero fre-quency is the particle mass, χ (0) = M .In conclusion, applying the fluctuation-dissipation relation we demonstrate that Eq. (10)is valid for arbitrary quantum systems describedby the Hamiltonian (7)-(9) and being at the ther-modynamic equilibrium state. The probabilitydistribution is of the form (16), where the suscep-tibility χ jj ( ω ) is the Fourier transform of the re-tarded thermodynamic Green function (14). Theformula (10) can be called the energy partition theorem for quantum systems because: (i) it isuniversal; (ii) it is an extension of the formulafor classical systems; (iii) it reduces to the energy equipartition theorem for high temperatures. Acknowledgement – The author would like tothank P. Hänggi and G.-L. Ingold for insightfuldiscussions on various aspects of this work andP. Talkner for suggestions regarding the proof ofnormalization. The work supported by the GrantNCN 2015/19/B/ST2/02856.
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