Quantum Counterpart of Equipartition Theorem in Quadratic Systems

The equipartition theorem is a fundamental law of classical statistical physics, which states that every degree of freedom contributes $k_{B}T/2$ to the energy, where $T$ is the temperature and $k_{B}$ is the Boltzmann constant. Recent studies have revealed the existence of a quantum version of the equipartition theorem. In the present work,we focus on how to obtain the quantum counterpart of the generalized equipartition theorem for arbitrary quadratic systems in which the multimode Brownian ocillators interact with multiple reservoirs at the same temperature. An alternative method of deriving the energy of the system is also discussed and compared with the result of the the quantum version of the equipartition theorem, after which we conclude that the latter is more reasonable. Our results can be viewed as an indispensable generalization of rencent works on a quantum version of the equipartition theorem.


I. INTRODUCTION
One of the elegant principles of classical statistical physics is the equipartition theorem, which has numerous applications in various topics, such as thermodynamics [1][2][3], astrophysics [4][5][6] and applied physics [7][8][9].It is natural to consider the quantum version of the equipartition theorem, since quantum mechanics has been founded over a hundred years and so as quantum statistical mechanics.
Recent years have seen much progress on this topic.A novel work [10] investigated the simplest quantum Brownian oscillator model to formulate the energy of the system in terms of the averge energy of a quantum oscillator in a harmonic well.Based on this work, more papers [11][12][13][14][15][16][17][18] tried to study quantum counterparts of the equipartition theorem in different versions of quadratic open quantum systems from various perspectives, including electrical circuits [11], dissipative diamagnetism [18] and focusing on kinetic energy for a more general setup [19].
In this work, we aim to deduce a quantum counterpart of the generalized equipartition theorem [20] for arbitray open quantum quadratic systems.Many quadratic systems share the same algebra [ Â, B] = iℏ, where the binary operator pair can be the coordinate x and the momentum p for oscillators, the magnetic flux Φ and charge Q for quantum circuits, and so on.We here turn to Brownian oscillators as an example to grasp the physical nature of all these systems.
To construct such systems, we adopt a generalized Calderia-Leggett model [21] and manage to transform it into a multi-mode Brownian-oscillator system welldiscussed in Ref. [22].For generality, we do not choose a concrete form of dissipation.We also generalize a formula in Ref. [23] for the internal energy so that it could be applied to the multi-mode Brownian oscillator system.It has been debated [24] which of the formula in Ref. [23]    * xinhai@iis.u-tokyo.ac.jp or the quantum counterpart of the equipartition theorem given in Ref. [10] truly describes the energy of the system.Our analysis shows that the generalized version of the former formula cannot be used to find the energy, which implies that the latter one is more reasonable.
The remainder of this paper is organized as follows.In Sec.II, we construct a quadratic system from the Calderia-Leggett model.In Sec.III we deduce the generalized equipartition theorem for this system and show its link to the conventioal equipartition theorem.In Sec.IV we try to give a multi-mode version of the remarkable formula in Ref. [23].More theoretical details are given in Appendixes.Numerical results are demonstrated in Sec.V. We summarize this paper in Sec.VI.Through out this paper, we set ℏ = 1 and β = 1/(k B T ) with k B being the Boltzmann constant and T being the temperature of the reservoirs if there is no special reminder.

II. ARBITRARY QUADRATIC SYSTEM
To construct our arbitrary quadratic system, let us start with the multi-mode Calderia-Leggett model [21] where u, v ∈ {1, 2, ..., n S } and j ∈ {1, 2, ..., n α } are indices for the oscillators in the system and in the αth bath, repectively.The coefficient c αuj represents the coupling strength between the coordinate of the uth oscillator in the system and the jth oscillator in the αth bath.The convention would put −c αuj in the last term of Eq. ( 1), but we replace it by c αuj .We also have the commutation relations for all the momentum and position operators as follows: arXiv:2309.14582v4 [cond-mat.stat-mech]14 Sep 2024 with δ representing the Kronecker delta.Equation ( 1) can be reorganized in the following forms: Here, the system-bath interaction results from the linear coupling of the system coordinate Qu and the random force Fαu .We also emphasize that all the mutually independent baths {h αB } in Eq. (3d) are at the same inverse temperature β.By defining the pure bath response function as we recognize that where F B αu (t) ≡ e ihBt Fαu e −ihBt and the average is defined over the canonical ensembles of baths as in ]/ α tr B (e −βhαB ).In Eq. ( 5) we use tilde to denote the Laplace transform f (ω) = ∞ 0 dω e iωt f (t) for any function f (t).By denoting V uv ≡ k uv + α φαuv (0) and Ω u ≡ M −1 u for convenience, we rewrite Eq. (3b) in the form which is the starting point of our quantum counterpart of the equipartition theorem.Here, H ren denotes the renormalization energy.The system Hamiltonian, referred to as Eq. ( 6), now is identical to the one presented in Ref. [22].Physically, we need V = {V uv }, k = {k uv } and Ω = {Ω u δ uv } to be positive definite.Without loss of generality, we can always set V and k to be symmetric.Detailed derivation of Eq. ( 5) can be found in Appendix A.

III. QUANTUM COUNTERPART OF GENERALIZAD EQUIPARTITION THEOREM
The conventional quantum counterpart of the equipartition theorem for the single-mode Calderia-Leggett model deals with the kinetic energy E k (β) in the Gibbs state of the total system with the inverse temperature being β [10] in the form of Here, we temporarily add ℏ for later convenience and E k [f (ω)] denotes the expectation of a function f (ω) over the normalized distribution function P k (ω), which satisifies P k (ω) ≥ 0 and ∞ 0 dω P k (ω) = 1.Equation ( 7) can be reduced to the classical case since lim However, when some degrees of freedom are interwined with each other, such as in our model [cf.Eq. ( 3)], we would better use the generalized the equipartition theorem [20].
In the rest of this work we study the quantity ⟨ Xi ∂H BO /∂ Xj ⟩ for any system degrees of freedom Xi , Xj ∈ { Pu }∪{ Qu }, with the average defined in the total Gibbs state ⟨...⟩ := tr T [...e −βHCL ]/ tr T e −βHCL , which is well defined since we assume that all the bath are at the same inverse temperature β.The derivative of the operator here is merely a notation, indicating that we initially treat all distinct operators as mutually independent variables like real numbers.After obtaining the result, We restore these variables back to operators.In the main text we focus on the cases Xi = Xj , while other cases are discussed in Appendix C.
We have Under the help of the fluctuation-disspation theorem [25] for symmetrized correlation function [24], we obtain with J QQ (ω) = {J QQ uv (ω)} being the anti-Hermitian part of the matrix χQQ (ω) = { χQQ uv (ω)} and {•, •} representing the anticommutator.Here, we denote the system response function of any two operators Âu and Bv as χ AB uv (t) ≡ i⟨[ Âu (t), Bv (0)]⟩.According to Ref. [22], we also have some useful relations for the quantities like χAB (ω).We list them below for later convenience: where V and Ω are given below Eq. ( 6).Equation ( 8) can be recast as with where we used the fact that J QQ (ω) is symmetric since χQQ (ω) is symmetric and therefore J QQ (ω) is real.
A similar process for the case Xi = Xj = Pu leads to Using the fluctuationdissipation theorem [25] again, we find where J P P (ω) = {J P P uv (ω)} is the anti-Hermitian part of matrix χP P (ω) = { χP P uv (ω)} [cf.Eq. ( 9b)].By substituting Eq. (9b) into Eq.( 12), we obtain the final result with The proof of positivity and normalization of Eqs. ( 11) and ( 14) can be found in Appendix B, by which we can recast Eqs. ( 10) and (13) as with where we use the notation P ii rather than P XiXi for simplicity.In Appendix C, we extend Eqs. ( 15) and ( 16) to the cases where Xi ̸ = Xj and summarize all the results as for any system degrees of freedom Xi , Xj ∈ { Pu } ∪ { Qu } with and E ij [...] denotes the expectation over P ij (ω).Here, we temporarily add ℏ for later convenience and let δ denote the Kronecker delta.Equations ( 17) and ( 18) are partly the main results of the present work.Discussions are presented here to conclude this section.Once we take the classical limit ℏ → 0 and the weak-coupling limit c αuj → 0, Eq. ( 17) reduces to ⟨X i ∂H S /∂X j ⟩ = δ ij /β [cf.Eq. ( 6)], which is termed as the generalized equipartition theorem [20].We also emphasize that though the right-hand side of Eq. ( 17) depends on different degrees of freedom (i and j), the function (ω/2) coth(βω/2) is universal for all the degrees of freedom, which is the "equipartition" in a quantum sense.Therefore, Eq. ( 17) is termed as the quantum counterpart of the generalized equipartition theorem.It is evident that Eqs.(11) and ( 14) reduce to the results in Refs .[10,24] for the single mode n S = 1 case.Besides, by noticing that ⟨ Pu ∂H BO /∂ Pu ⟩ equals twice the kinetic energy of the uth oscillator and J P P (ω) = ωΩ −1 J QQ (ω)Ω −1 [cf.Eq. (9b)], Eqs. ( 13) and ( 14) reduce to the results presented in Ref. [19].
Equation (17) here offers a new angle on how to calculate the quantities of open systems, which is generally hard to obtain.An application is given below.Noting that the total energy is given by E with E[...] denotes the expectation over which is checked [cf.Eq. ( 16)] to be nonnegative and normalized over R + .Equation ( 19) is termed as the quantum counterpart of conventional equipartition theorem [24].Moving further with the help of thermodynamic equations, we can determine the free energy F (β) of the system by and hence Eqs. ( 21) and ( 19) may yield from which we further obtain an expression for the partition function of the system in the form of Note that Eq. ( 23) is much easier to obtain than the conventional influence-functional approach [26,27].

IV. ALTERNATIVE APPROACH FOR THE ENERGY
A recent review [24] presented another approach to obtain the energy of the system of multi-mode harmonic oscillators.When introduced in Ref. [23] first, the result was only limited to the single-mode case.Here we generalize their derivation and find that their derivation is not applicable to the multi-mode case.
The starting point of Ref. [23] is quite straightforward.Since the conventional definition for the internal energy of the oscillator U S (β) = tr T [H BO e −βHCL ]/ tr e −βHCL is generally challenging to handle, we adopt a normal-mode coordinates, so that the transformed Hamiltonian H T describes N (= 1+ α n α ) uncoupled oscillators.Physically we do not need to obtain the detailed information for any normal modes, since the total energy U T (β) is only associated with the normal frequencies, namely with ω r being the normal frequency for r-th oscillator in the transformed system.Here, we also introduced the notation u(ω, β) ≡ (ω/2) coth(βω/2) for later convenience.
Since the energy for the independent bath is well-defined as the authors of Ref. [23] interpreted the difference as the internal energy and found it to be where χQQ (ω) is the one-dimensional version of Eq. (9a).
Following their procedures for the multi-mode case, we find (see Appendix D for detailed derivation) with which does not give us U S (β) = U T (β) − U B (β).On the other hand, the result of the system internal energy according to the quantum counterpart of equipartition theorem [cf.Eq. ( 19)] is applicable to any multi-mode case.Therefore, we conclude that Eq. ( 19) is more reasonable than the alternative approach dicussed in Ref. [23] as an expression for the internal energy of the system.Equations ( 28) and ( 29) are another part of the main results of this work.

V. NUMERICAL DEMONSTRATIONS
In this section, we use the two-mode (u, v ∈ {1, 2}) Brownian-oscillator system coupled with one reservoir (α = 1) to give a numerical demonstration of our results.The system Hamiltonian H S of the two-mode system reads 2 + 2V 12 Q1 Q2 )/2, while the system-bath interaction term becomes H SB = Q1 F1 + Q2 F2 with the random force Fu1 = j c uj xj for u ∈ {1, 2}.The bath Hamiltonian reduces to h B = j (p 2 j /2m j + m j ω 2 j x2 j /2).To enhance clarity, we choose the spectrum of the pure bath in the following form: with η ≡ {η uv = η u δ uv } specifying the strength of the system-bath couplings.From an experimental point of view, this setup can be realized, for example, in molecular junctions [28][29][30].We also introduce the parameter λ ∈ {1, 1.25, 1.5} to vary the strength, which can be realized experimentally by modifying the intermolecular distance.
Through out this section, we select the parameters in the unit of Ω B as γ The strength of the couplings are chosen to be η 1 = 0.2Ω B and η 2 = 0.1Ω B .Figures (1a) and (1b) depict P(ω) and B(ω) in the three cases.As λ decreases, the curves become sharper around the square root of the eigenvalues of ΩV .In other words, the maximum points of P(ω) and B(ω) become closer to them as λ decreases.In fact, we can prove the following (see Appendix E) where the summation is over all the square root of eigenvalues of the matrix ΩV (considering multiplicity).These results show that in the weak-coupling limit, only the oscillators with these typical frequencies contribute to the quantity that we consider, such as the energy.In this case, the energy reads which is how the quantum counterpart of equipartition theorem behaves in the weak-coupling limit.In the single-mode case, a similar pattern has also been discussed in Ref. [24] VI.SUMMARY To summarize, we derived a quantum counterpart of the generalized equipartition for arbitray quadratic systems, which we can also reduce to the results presented in previous works for the single-mode case.We also extended another formula for the internal energy of the multi-mode Brownian-oscillator system.The generalized formula as well as our analysis shed light on the controversies upon the method.We noticed that our quantum counterpart of equipartition theorem can be used to obtain the partition function of the system in a much easier way than the classical approach [27].Our results can be viewed as an indispensable development of rencent works on the quantum counterpart of the equipartition theorem.
As a future prospect, expressing thermodynamic quantities as an infinite series also offers potential advantages for this objective [24].Work in this direction is in progress.As another point, it seems difficult to discuss the quantum version of equipartition theorem without the help of fluctuation-disspation theorem and to con-sider it over steady states or even in general nonequilibrium.Discussing the present topic under other more difficult setups, such as quartic systems, is also challenging.All of them constitute directions of further development.

Appendix D: Alternative approach for the energy
To obtain Eqs. ( 28) and ( 29) for the multi-mode case, we set N = n S + α n α and follow the procedures in Ref. [23]: (I) From a normal-mode analysis we obtain the following relation bewteen χQQ (ω) and all the normal frequencies {ω r }: whose derivation is presented hereafter.Applying a similar treatment as in Appendix A to Qu (t) and Pu (t), we obtain where ω is the normal frequency.To perform a normalmode analysis, we put Eq. (D3) into Eqs.(A2) and (D2), finding Solving Eq. (D4b) for x αj (ω), and substituting it into Eq.(D4a), we obtain To obtain nontrivial normal frequencies, we require [cf.Eq. (A5)] which is also an equation for all the normal frequencies {ω}.As a function with respect to ω 2 [cf.Eqs.(9a) and (A5)], det χQQ (ω) has singular points at all {ω 2 r } while zero point at all {ω 2 αj }.Therefore we can write det χQQ (ω) in the form of Eq. (D1).Note that n S is the dimension of the matrix and the factor (−1) nS comes from the change of sign in the determinant.
(II) Once we denote A(z) ≡ det χQQ (z 1/2 ), mathematically it is easy to know which helps us to recast Eq. ( 24) as By the residue theorem we further write Eq. (D8) as where the contour C 1 is shown in Fig. 2a.Therefore we obtain Eqs. ( 28) and (29), which reduce to Eq. ( 26) when n S = 1.

FIG. 1 :
FIG. 1: Plots of P(ω) in (a) and B(ω) in (b) when λ ∈ {1, 1.25, 1.5}.Here the red dots on the horizontal axes represent ω = 0.7071Ω B and ω = 1.224ΩB , respectively, which are the square root of eigenvalues of the matrix ΩV according to the parameters chosen in the main text.