Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Derivation of the density operator with quantum analysis forthe generalized Gibbs ensemble in quantum statistics
Masamichi Ishihara ∗ Department of Human Life Studies, Koriyama Women’s University, Koriyama, Fukushima, 963-8503,JAPAN
Abstract
We derived the equation of the density operator for generalized entropy and generalizedexpectation value with quantum analysis when conserved quantities exist. The derivedequation is simplified when the conventional expectation value is employed. The derivedequation is also simplified when the commutation relations, [ˆ ρ, ˆ H ] and [ˆ ρ, ˆ Q [ a ] ], are thefunctions of the density operator ˆ ρ , where ˆ H is the Hamiltonian, and ˆ Q [ a ] is the conservedquantity. We derived the density operators for the von Neumann entropy, the Tsallis en-tropy, and the R´enyi entropy in the case of the conventional expectation value. We alsoderived the density operators for the Tsallis entropy and the R´enyi entropy in the case ofthe escort average (the normalized q -expectation value), when the density operator com-mutes with the Hamiltonian and the conserved quantities. We found that the argumentof the density operator for the canonical ensemble is simply extended to the argumentfor the generalized Gibbs ensemble in the case of the conventional expectation value,even when conserved quantities do not commute. The simple extension of the argumentis also shown in the case of the escort average, when the density operator ˆ ρ commuteswith the Hamiltonian ˆ H and the conserved quantity ˆ Q [ a ] : [ˆ ρ, ˆ H ] = [ˆ ρ, ˆ Q [ a ] ] = 0. Thesefindings imply that the argument of the density operator for the canonical ensemble issimply extended to the argument for the generalized Gibbs ensemble in some systems. Keywords: density operator, quantum analysis, generalized Gibbs ensemble, quantumstatistics, nonextensive statistics
1. Introduction
The density operator is widely used to calculate physical quantities in various systems.Therefore, it is required to determine the density operator with adequate constraints. Amethod to obtain the density operator with constraints is the maximum entropy principle(MEP). The maximum entropy principle is useful to determine the density operator underconstraints when an adequate entropy is given. The density operators for some entropies ∗ Corresponding author. Tel.: +81 24 932 4848; Fax: +81 24 933 6748.
Email address: [email protected] (Masamichi Ishihara)
Preprint submitted to Elsevier July 8, 2020 ere derived in the MEP [1]. A density operator is applied to calculate values of physicalquantities.The generalized Gibbs ensemble (GGE) [2–4] is an extension of the canonical ensem-ble, and some conserved quantities are contained in the density operator for the GGE.The density operator for the GGE was derived in the MEP [3, 5], and it is shown that theargument of the density operator is simply extended for the von Neumann entropy withthe conventional expectation value. It is considered that some systems can be describedby the GGE.A nonextensive statistics is an extension of the Boltzmann-Gibbs statistics, and manyentropies have been proposed. For example, the Tsallis [6–9] and the R´enyi entropies [8–10] are well-known entropies. A difference between the Boltzmann-Gibbs statistics andthe unconventional statistics is the definition of the expectation value. The escort average(the normalized q -expectation value) is often used in the statistics. The density operatorcan be obtained in the MEP, and is applied to various phenomena. The constraintof energy conservation is imposed in such calculations. It is natural to consider someadditional constraints of conserved quantities.The quantum analysis developed by M. Suzuki [11–16] is a useful tool to obtain thedensity operator in the MEP. The quantum analysis was applied to the nonequilibriumresponse [14] and quantum correlation identities [15]. The density operators for someentropies were derived with the quantum analysis in the previous study [17]. It is expectedthat the quantum analysis works well in order to obtain the density operators with someconstraints in the MEP.In this paper, we attempt to obtain the density operators with the quantum analysisin the MEP. Some entropies such as Tsallis and R´enyi entropies are extremized under theexistence of conserved quantities. Density operators are explicitly obtained in some cases,and conditions to obtain the well-known density operators for the GGE are explicitlyshown. The density operators in some cases are given with the quantum analysis in theMEP.We derived the equation of the density operator for generalized entropy and general-ized expectation value when conserved quantities exist. It was shown that the equationof the density operator is simplified in particular cases, and the density operators forthe von Neumann, Tsallis and R´enyi entropies were obtained. We found the followings.The argument of the density operator for the canonical ensemble is simply extended tothe argument for the GGE without assuming commutation relations between conservedquantities, when the conventional expectation value is employed. The extension is possi-ble for the unconventional expectation value, when the density operator commutes withthe Hamiltonian and the conserved quantities. The simple extensions of the densityoperator for the GGE are shown in quantum statistics.This paper is organized as follows. In section 2, the quantum analysis is briefly re-viewed, and the variations of some functionals are calculated. In section 3, the equationof the density operator is derived in the MEP. The density operators for the von Neu-mann, Tsallis, and R´enyi entropies are derived when conserved quantities exist. The lastsection is assigned for discussion and conclusion.2 . Quantum analysis and variations of functionals We begin with the following differential df ( ˆ A ) with operators ˆ A and d ˆ A : df ( ˆ A ) = lim ε → f ( ˆ A + εd ˆ A ) − f ( ˆ A ) ε . (1)The operators ˆ A and d ˆ A do not commute generally. Symbolically, the above differentialis represented as df ( ˆ A ) = df ( ˆ A ) d ˆ A d ˆ A. (2)The hyperoperator df ( ˆ A ) /d ˆ A maps d ˆ A to df ( ˆ A ). This hyperoperator df ( ˆ A ) /d ˆ A is namedquantum derivative in the quantum analysis. A hyperoperator ˆ L ˆ A is defined as the leftmultiplication to the operator: ˆ L ˆ A ˆ B := ˆ A ˆ B. (3)The inner derivative ˆ δ ˆ A is defined by using the commutation relation:ˆ δ ˆ A ˆ B := [ ˆ A, ˆ B ] = ˆ A ˆ B − ˆ B ˆ A. (4)The hyperoperator ˆ L ˆ A and the inner derivative ˆ δ ˆ A commute:ˆ L ˆ A ˆ δ ˆ A ˆ B = ˆ δ ˆ A ˆ L ˆ A ˆ B. (5)This property is useful in calculations. Hereafter, we do not distinguish between ˆ L ˆ A andˆ A throughout this paper as in Ref. [11]The quantum derivative df ( ˆ A ) /d ˆ A is represented as follows: df ( ˆ A ) d ˆ A = Z dtf (1) ( ˆ A − t ˆ δ ˆ A ) , (6)where f ( k ) ( x ) is the k -th derivative of f ( x ) with respect to a classical variable x . The firstorder quantum Taylor expansion for operators ˆ A and ˆ B is represented with df ( ˆ A ) /d ˆ A : f ( ˆ A + x ˆ B ) = f ( ˆ A ) + x df ( ˆ A ) d ˆ A ˆ B + O ( x ) . (7)The variation of the functional is often used. The variation of the functional F [ A ( t )]of A ( t ) with a variable t [13, 16] is defined as follows: δF [ ˆ A ( t )] := lim ε → F [ ˆ A ( t ) + ǫ ( δ ˆ A ( t ))] − F [ ˆ A ( t )] ε . (8)These expressions are used to derive the density operator in the MEP. We attemptto obtain the expressions of variations for some functionals in the next subsection.3 .2. Variations of functionals with quantum analysis In this subsection, we calculate variations of some functionals with the quantumanalysis. We treat following functionals: F (ˆ ρ ) = Tr( f (ˆ ρ )) , (9a) G (ˆ ρ ; ˆ A ) = Tr( g (ˆ ρ ) ˆ A )Tr( g (ˆ ρ )) , (9b) H (ˆ ρ ) = h ( F (ˆ ρ )) ≡ h (Tr( f (ˆ ρ ))) . (9c)First, we treat the functional F (ˆ ρ ). The variation of F (ˆ ρ ) is obtained by applying thequantum Taylor expansion, Eq. (7): δF (ˆ ρ ) = lim ε → ε − (cid:26) Tr[ f (ˆ ρ + ε ( δ ˆ ρ ))] − Tr[ f (ˆ ρ )] (cid:27) = lim ε → ε − (cid:26) Tr (cid:20) f (ˆ ρ ) + df (ˆ ρ ) d ˆ ρ ε ( δ ˆ ρ ) + O ( ε ) (cid:21) − Tr[ f (ˆ ρ )] (cid:27) = Tr (cid:20) df (ˆ ρ ) d ˆ ρ ( δ ˆ ρ ) (cid:21) . (10)Applying Eq. (6) to the above equation, we obtain δF (ˆ ρ ) = Tr (cid:20)Z dtf (1) (ˆ ρ − t ˆ δ ˆ ρ )( δ ˆ ρ ) (cid:21) = Tr "Z dt ∞ X k =0 k ! f ( k +1) (ˆ ρ )( − t ) k (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1) = ∞ X k =0 ( − k ( k + 1)! Tr h f ( k +1) (ˆ ρ ) (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1)i . (11)We focus on the trace in Eq. (11) for k ≥
1. The term ((ˆ δ ˆ ρ ) k ( δ ˆ ρ )) generates 2 k termsof the form ˆ ρ j ( δ ˆ ρ )ˆ ρ k − j (0 ≤ j ≤ k ) . This term gives the same contribution when thecyclic permutation property of trace is hold:Tr h f ( k +1) (ˆ ρ )ˆ ρ j ( δ ˆ ρ )ˆ ρ k − j i = Tr h ˆ ρ k − j f ( k +1) (ˆ ρ )ˆ ρ j ( δ ˆ ρ ) i = Tr h f ( k +1) (ˆ ρ )ˆ ρ k ( δ ˆ ρ ) i . (12)The number of the terms with plus sign equals that with minus sign. Therefore, weobtain Tr h f ( k +1) (ˆ ρ ) (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1)i = 0 ( k ≥ . (13)Equation (13) is also proved with the result given in Appendix A. Equation (13) gives δF (ˆ ρ ) = Tr h f (1) (ˆ ρ )( δ ˆ ρ ) i . (14)Next, we attempt to calculate δG (ˆ ρ ; ˆ A ). We pay attention that ˆ ρ and ˆ A do notcommute generally. In the same way, we obtain the expression of δG (ˆ ρ ; ˆ A ) by applying4qs. (6) and (7). δG (ˆ ρ ; ˆ A ) = 1Tr( g (ˆ ρ )) " ∞ X k =0 ( − ) k ( k + 1)! Tr (cid:16) g ( k +1) (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:17) ˆ A (cid:17) − Tr( g (ˆ ρ ) ˆ A )Tr( g (ˆ ρ )) 1Tr( g (ˆ ρ )) " Tr (cid:16) g (1) (ˆ ρ )( δ ˆ ρ ) (cid:17) . (15)Equation (15) is simplified when the conventional expectation value is employed: thefunction g ( x ) is cx , where c is a constant. Therefore, we have g (1) ( x ) = c and g ( j ) ( x ) = 0( j ≥ δG (ˆ ρ ; ˆ A ) = Tr(( δ ˆ ρ ) ˆ A )Tr(ˆ ρ ) − Tr(ˆ ρ ˆ A )Tr(ˆ ρ ) Tr( δ ˆ ρ )Tr(ˆ ρ ) for g (ˆ ρ ) = c ˆ ρ. (16)We note that the commutation relation between ˆ ρ and ˆ A is not assumed to obtainEq. (16). Equation (15) is also simplified when the commutation relation between ˆ ρ andˆ A is a function of ˆ ρ : [ˆ ρ, ˆ A ] = r (ˆ ρ ). The trace, Tr( u (ˆ ρ )((ˆ δ ˆ ρ ) k ( δ ˆ ρ )) ˆ A ), isTr( u (ˆ ρ )((ˆ δ ˆ ρ ) k ( δ ˆ ρ )) ˆ A ) = k ≥ − Tr( u (ˆ ρ ) r (ˆ ρ )( δ ˆ ρ )) ( k = 1)Tr( u (ˆ ρ )( δ ˆ ρ ) ˆ A ) ( k = 0) , (17)where u (ˆ ρ ) is a function of ˆ ρ (see Appendix B). We have δG (ˆ ρ ; ˆ A ) = 1Tr( g (ˆ ρ )) Tr nh ( ˆ A − G (ˆ ρ ; ˆ A )) g (1) (ˆ ρ ) + 12 g (2) (ˆ ρ ) r (ˆ ρ ) i ( δ ˆ ρ ) o for [ˆ ρ, ˆ A ] = r (ˆ ρ ) . (18)Finally, we treat H (ˆ ρ ). The variation δH (ˆ ρ ) is easily obtained by using the expressionof δF (ˆ ρ ). δH (ˆ ρ ) = lim ε → H (ˆ ρ + ε ( δ ˆ ρ )) − H (ˆ ρ ) ε = lim ε → h ( F (ˆ ρ ) + εδF (ˆ ρ ) + O ( ε )) − h ( F (ˆ ρ )) ε = lim ε → [ h ( F (ˆ ρ )) + h (1) ( F (ˆ ρ ))( εδF (ˆ ρ ) + O ( ε )) + O ( ε )] − h ( F (ˆ ρ )) ε = h (1) ( F (ˆ ρ ))( δF (ˆ ρ )) . (19)In the next section, these results are used to derive the density operators in the MEP.
3. Derivation of the density operators in the maximum entropy principle
We treat the entropy S (ˆ ρ ) under some constraints, where ˆ ρ is a density operator. Theexpectation value of a quantity ˆ A is denoted by h ˆ A i . The quantity ˆ Q [ a ] is the operatorof a conserved quantity. Therefore, the operator ˆ Q [ a ] and the Hamiltonian ˆ H commute:5 ˆ H, ˆ Q [ a ] ] = 0. The number of the constraints represented by the operators ˆ Q [ a ] is M .Therefore, we treat the variational problem with the following functional I (ˆ ρ ): I (ˆ ρ ) := S (ˆ ρ ) − α (Tr ˆ ρ − − β ( h ˆ H i − E ) − M X a =1 γ [ a ] ( h ˆ Q [ a ] i − Q [ a ] ) , (20)where α, β, γ [ a ] are Lagrange multipliers, E is the energy, and the expectation value ofthe physical quantity ˆ Q [ a ] is Q [ a ] . We note that the above expectation value is not alwaysthe conventional expectation value defined by Tr(ˆ ρ ˆ A ) / Tr(ˆ ρ ).The maximum entropy principle requires δI (ˆ ρ ) = 0. We treat the following entropyand average in this paper. S (ˆ ρ ) := H (ˆ ρ ) ≡ h (Tr( f (ˆ ρ ))) = h ( F (ˆ ρ )) , (21a) h ˆ Q [ a ] i := G (ˆ ρ ; ˆ Q [ a ] ) ≡ Tr( g (ˆ ρ ) ˆ Q [ a ] )Tr( g (ˆ ρ )) . (21b)The variation of the entropy δS (ˆ ρ ) is given by δS (ˆ ρ ) = h (1) ( F (ˆ ρ ))Tr (cid:0) f (1) (ˆ ρ )( δ ˆ ρ ) (cid:1) = Tr (cid:0) h (1) ( F (ˆ ρ )) f (1) (ˆ ρ )( δ ˆ ρ ) (cid:1) . (22)The functional I (ˆ ρ ) is rewritten with G (ˆ ρ, ˆ A ): I (ˆ ρ ) = S (ˆ ρ ) − α (Tr ˆ ρ − − β ( G (ˆ ρ ; ˆ H ) − E ) − M X a =1 γ [ a ] ( G (ˆ ρ ; ˆ Q [ a ] ) − Q [ a ] ) . (23)The variation of I (ˆ ρ ) is given by δI (ˆ ρ ) = δS (ˆ ρ ) − α (Tr( δ ˆ ρ )) − β ( δG (ˆ ρ ; ˆ H )) − M X a =1 γ [ a ] ( δG (ˆ ρ ; ˆ Q [ a ] )) . (24)With this expression, we attempt to find the density operators in the following subsection. The variation δI (ˆ ρ ), Eq. (24), is reduced to simple expressions in particular cases.In this subsection, we treat two cases: (i) the conventional expectation value is em-ployed, and (ii) the commutation relations, [ˆ ρ, ˆ H ] = r [0] (ˆ ρ ) and [ˆ ρ, ˆ Q [ a ] ] = r [ a ] (ˆ ρ ) ( a =1 , , · · · , M ), are satisfied. Especially, density operators are given for r [0] (ˆ ρ ) = r [ a ] (ˆ ρ ) = 0when the escort average is employed. The function g ( x ) is cx for the conventional expectation value, where c is a constant.This leads to g (1) = c and g ( k ) ( x ) = 0 ( k ≥ δI (ˆ ρ ) = Tr " h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) − α − β ( ˆ H − h ˆ H i )Tr(ˆ ρ ) ! − M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i )Tr(ˆ ρ ) !! ( δ ˆ ρ ) . (25)6he requirement δI (ˆ ρ ) = 0 leads to h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) − α − β ( ˆ H − h ˆ H i )Tr(ˆ ρ ) ! − M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i )Tr(ˆ ρ ) ! = 0 . (26)Considering the requirement Tr(ˆ ρ ) = 1, we have h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) = α + β ( ˆ H − h ˆ H i ) + M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) . (27)It is significant that Eq. (27) is derived without assuming the commutation relationsbetween operators: the relation between ˆ Q [ a ] and ˆ Q [ b ] is not assumed. Equation (27)indicates that the argument of the density operator for the canonical ensemble is simplyextended to the argument for the GGE.First, we treat the von Neumann entropy as an example. The function f ( x ) is − x ln x and h ( x ) is x for the von Neumann entropy. The derivatives, f (1) ( x ) = − ln x − h (1) ( x ) = 1, are obtained. Inserting these expressions into Eq. (27), we haveˆ ρ = exp (cid:16) − α − − β ( ˆ H − h ˆ H i ) − M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) (cid:17) . (28)The requirement Tr(ˆ ρ ) = 1 leads toˆ ρ = 1 Z convN exp − β ˆ H − M X a =1 γ [ a ] ˆ Q [ a ] ! , (29a) Z convN = Tr " exp − β ˆ H − M X a =1 γ [ a ] ˆ Q [ a ] ! . (29b)This is just the density operator for the GGE.Next, we treat the Tsallis entropy. The function f ( x ) is x q and h ( x ) is (1 − x ) / ( q − f (1) ( x ) = qx q − and h (1) ( x ) = 1 / (1 − q ), areinserted into Eq. (27): q − q ˆ ρ q − = α + β ( ˆ H − h ˆ H i ) + M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) . (30)With the requirement Tr(ˆ ρ ) = 1, we haveˆ ρ = 1 Z convT (cid:16) β ( ˆ H − h ˆ H i ) + M X a =1 ˜ γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i (cid:17) q − , (31a) Z convT = Tr h(cid:16) β ( ˆ H − h ˆ H i ) + M X a =1 ˜ γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i (cid:17) q − i , (31b)˜ β := β/α, (31c)˜ γ [ a ] := γ [ a ] /α. (31d)7e have the following relation by multiplying ˆ ρ and calculating the trace fromEq. (30): q − q Tr(ˆ ρ q ) = α. (32)Finally, we treat the R´enyi entropy. The function f ( x ) is x q and h ( x ) is ln x/ (1 − q )for the R´enyi entropy. The derivatives, f (1) = qx q − and h (1) ( x ) = 1 / ((1 − q ) x ), areinserted into Eq. (27): q − q ρ q ) ˆ ρ q − = α + β ( ˆ H − h ˆ H i ) + M X a =1 γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) . (33)With the requirement Tr(ˆ ρ ) = 1, we haveˆ ρ = 1 Z convR (cid:16) β ( ˆ H − h ˆ H i ) + M X a =1 ˜ γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i (cid:17) q − , (34a) Z convR = Tr h(cid:16) β ( ˆ H − h ˆ H i ) + M X a =1 ˜ γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i (cid:17) q − i , (34b)˜ β := β/α, (34c)˜ γ [ a ] := γ [ a ] /α. (34d)We have the following relation by multiplying ˆ ρ and calculating the trace from Eq. (33): q − q = α. (35)The density operator for the canonical ensemble is simply extended to the densityoperator for the GGE in the conventional expectation value case. [ˆ ρ, ˆ H ] = r [0] (ˆ ρ ) and [ˆ ρ, ˆ Q [ a ] ] = r [ a ] (ˆ ρ ) ( a = 1 , , · · · , M )We treat the cases of [ˆ ρ, ˆ H ] = r [0] (ˆ ρ ) and [ˆ ρ, ˆ Q [ a ] ] = r [ a ] (ˆ ρ ) ( a = 1 , , · · · , M ), where r [0] (ˆ ρ ) and r [ a ] (ˆ ρ ) are functions of ˆ ρ . The cases contain the following conditions: [ˆ ρ, ˆ H ] =0 and [ˆ ρ, ˆ Q [ a ] ] = 0 ( a = 1 , , · · · , M ). The density operator constructed from ˆ H andˆ Q [ a ] satisfies [ˆ ρ, ˆ H ] = [ˆ ρ, ˆ Q [ a ] ] = 0 when the commutation relations [ ˆ Q [ a ] , ˆ Q [ b ] ] = 0( a, b = 1 , , · · · , M ) are satisfied.The variation δI (ˆ ρ ) is given by using Eqs. (22) and (18) in the case of [ˆ ρ, ˆ H ] = r [0] (ˆ ρ )and [ˆ ρ, ˆ Q [ a ] ] = r [ a ] (ˆ ρ ) ( a = 1 , , · · · , M ): δI (ˆ ρ ) =Tr (" h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) − α − β Tr( g (ˆ ρ )) (cid:16)(cid:0) ˆ H − h ˆ H i (cid:1) g (1) (ˆ ρ ) + 12 g (2) (ˆ ρ ) r [0] (ˆ ρ ) (cid:17) − M X a =1 γ [ a ] Tr( g (ˆ ρ )) (cid:16)(cid:0) ˆ Q [ a ] − h ˆ Q [ a ] i (cid:1) g (1) (ˆ ρ ) + 12 g (2) (ˆ ρ ) r [ a ] (ˆ ρ ) (cid:17) ( δ ˆ ρ ) ) . (36)8herefore, the condition δI (ˆ ρ ) = 0 gives h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) − α − β Tr( g (ˆ ρ )) (cid:16)(cid:0) ˆ H − h ˆ H i (cid:1) g (1) (ˆ ρ ) + 12 g (2) (ˆ ρ ) r [0] (ˆ ρ ) (cid:17) − M X a =1 γ [ a ] Tr( g (ˆ ρ )) (cid:16)(cid:0) ˆ Q [ a ] − h ˆ Q [ a ] i (cid:1) g (1) (ˆ ρ ) + 12 g (2) (ˆ ρ ) r [ a ] (ˆ ρ ) (cid:17) = 0 . (37)We treat the cases of [ˆ ρ, ˆ H ] = [ˆ ρ, ˆ Q [ a ] ] = 0 ( a = 1 , , · · · , M ) hereafter. The require-ment δI (ˆ ρ ) = 0 for [ˆ ρ, ˆ H ] = [ˆ ρ, ˆ Q [ a ] ] = 0 gives h (1) ( F (ˆ ρ )) f (1) (ˆ ρ ) − α − β Tr( g (ˆ ρ )) (cid:0) ˆ H − h ˆ H i (cid:1) g (1) (ˆ ρ ) − M X a =1 γ [ a ] Tr( g (ˆ ρ )) (cid:0) ˆ Q [ a ] − h ˆ Q [ a ] i (cid:1) g (1) (ˆ ρ ) = 0 . (38)With Eq. (38), we attempt to find the density operators for the Tsallis and R´enyi en-tropies with the escort average.We attempt to obtain the density operator for the Tsallis entropy with the escortaverage. The function f ( x ) is x q , h ( x ) is (1 − x ) / ( q −
1) for the Tsallis entropy, andthe function g ( x ) is x q for the escort average. Inserting these functions into Eq. (38), wehave q − q ! ˆ ρ q − − α − qβ ( ˆ H − h ˆ H i )Tr(ˆ ρ q ) ! ˆ ρ q − − M X a =1 qγ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i )Tr(ˆ ρ q ) ! ˆ ρ q − = 0 . (39)Multiplying ˆ ρ and taking the trace, we obtain q − q Trˆ ρ q − α = 0 . (40)We have the density operator for the Tsallis entropy with the escort average with therequirement Trˆ ρ = 1.ˆ ρ = 1 Z escT − (1 − q ) β ( ˆ H − h ˆ H i )Tr(ˆ ρ q ) ! − M X a =1 (1 − q ) γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i )Tr(ˆ ρ q ) !! − q , (41a) Z escT = Tr " − (1 − q ) β ( ˆ H − h ˆ H i )Tr(ˆ ρ q ) ! − M X a =1 (1 − q ) γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i )Tr(ˆ ρ q ) !! − q . (41b)Equation (41a) is the density operator for the GGE for the Tsallis entropy with theescort average. The conditions [ˆ ρ, ˆ Q [ a ] ] = 0 ( a = 1 , , · · · , M ) are satisfied in the case of[ ˆ Q [ a ] , ˆ Q [ b ] ] = 0. Therefore, Eq. (41a) gives the density operator for the Tsallis entropywith the escort average when the conserved quantities commute: [ ˆ Q [ a ] , ˆ Q [ b ] ] = 0.9nother example is the R´enyi entropy with the escort average. For the R´enyi entropy,the function f ( x ) is x q , g ( x ) is x q , and h ( x ) is (ln x ) / (1 − q ). These functions yield (cid:18) q − q ρ q ) (cid:19) ˆ ρ q − − α − qβ ( ˆ H − h ˆ H i )Tr(ˆ ρ q ) ! ˆ ρ q − − M X a =1 qγ [ a ] ˆ Q [ a ] − h ˆ Q [ a ] i Tr(ˆ ρ q ) ! ˆ ρ q − = 0 . (42)Multiplying ˆ ρ and taking the trace, we obtain (cid:18) q − q (cid:19) − α = 0 . (43)We have the density operator for the R´enyi entropy with the escort average with therequirement Trˆ ρ = 1.ˆ ρ = 1 Z escR − (1 − q ) β ( ˆ H − h ˆ H i ) − M X a =1 (1 − q ) γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) ! − q , (44a) Z escR = Tr " − (1 − q ) β ( ˆ H − h ˆ H i ) − M X a =1 (1 − q ) γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) ! − q . (44b)Equation (44a) is the density operator for the GGE for the R´enyi entropy with the escortaverage. The conditions [ˆ ρ, ˆ Q [ a ] ] = 0 ( a = 1 , , · · · , M ) are satisfied for [ ˆ Q [ a ] , ˆ Q [ b ] ] = 0.Eq. (44a) gives the density operator for the R´enyi entropy with the escort average whenthe conserved quantities commute: [ ˆ Q [ a ] , ˆ Q [ b ] ] = 0. The density operator for the R´enyientropy is similar to the density operator for the Tsallis entropy.
4. Discussion and Conclusion
We derived the equation of the density operator in the maximum entropy principleby using the quantum analysis. The derived equation is simplified in particular cases,and the density operators for the generalized Gibbs ensemble are derived: the densityoperators for the von Neumann entropy, the Tsallis entropy, and the R´enyi entropy inthe case of the conventional expectation value and the density operators for the Tsallisentropy and the R´enyi entropy in the case of the escort average (the normalized q -expectation value). The obtained density operators are the simple extensions of thewell-known density operators for the canonical ensemble.The derived equation of the density operator is simplified in two cases in the presentpaper. The equation is simplified when the conventional expectation value is employedand the equation is also simplified when the following commutation relations are satisfied:[ˆ ρ, ˆ H ] = r [0] (ˆ ρ ) and [ˆ ρ, ˆ Q [ a ] ] = r [ a ] (ˆ ρ ) ( a = 1 , , · · · , M ), where ˆ ρ is the density operator,ˆ H is the Hamiltonian, ˆ Q [ a ] is the conserved quantity, and r [ j ] (ˆ ρ ) is a function of ˆ ρ ( j = 0 , , , · · · , M ).The argument of the density operator for the generalized Gibbs ensemble is givenby adding the conserved quantities to the argument of the density operator for thecanonical ensemble when the conventional expectation value is employed: the argument10 β ( ˆ H − h ˆ H i ) in the canonical ensemble is replaced with the argument − β ( ˆ H − h ˆ H i ) − X a γ [ a ] ( ˆ Q [ a ] − h ˆ Q [ a ] i ) in the generalized Gibbs ensemble. There is no assumption of thecommutation relations between the conserved quantities, while the Hamiltonian and theconserved quantity commute.The argument of the density operator for the canonical ensemble is extended simplyto the argument of the density operator for the generalized Gibbs ensemble in the caseof the escort average (the normalized q -expectation value), as found in the case of theconventional expectation value, when the density operator ˆ ρ , the Hamiltonian ˆ H , andconserved quantities ˆ Q [ a ] ( a = 1 , , · · · , M ) satisfy the following commutation relations:[ˆ ρ, ˆ H ] = 0 and [ˆ ρ, ˆ Q [ a ] ] = 0. We note that the conditions, [ˆ ρ, ˆ H ] = 0 and [ˆ ρ, ˆ Q [ a ] ] = 0, aresatisfied when the commutation relations [ ˆ Q [ a ] , ˆ Q [ b ] ] = 0 ( a, b = 1 , , · · · , M ) are satisfied.The argument of the density operator is extended simply even when the expectation valueis not conventional.It is clearly shown for various entropies that the argument of the density operator forthe canonical ensemble is simply extended to the argument for the generalized Gibbs en-semble when the conventional expectation value is employed, even though the conservedquantities do not commute. This implies that the well-known density operator for thegeneralized Gibbs ensemble appears when there are conserved quantities. In contrast,the argument of the density operator for the canonical ensemble is not always extendedsimply in the case of the unconventional expectation value, when there are conservedquantities. The argument of the density operator may be simply extended in the case ofunconventional expectation value when additional conditions are imposed. Such condi-tions are given in the present paper: the density operator and the Hamiltonian commute,and the density operator and the conserved quantities commute. The conditions will besatisfied when the conserved quantities commute each other.We derived the density operators for the generalized Gibbs ensemble in the particularcases. It is not enough to study the derivation of the density operator in the cases ofunconventional expectation values, when the density operator and the conserved quantitydo not commute. The explicit commutation relations between the conserved quantitiesshould be required to derive the density operator. The derivation of the density operatorin such cases will be studied in the future. Appendix A. Expansion of ((ˆ δ ˆ ρ ) k ˆ A ) and proof of Tr( f ( ˆ ρ )((ˆ δ ˆ ρ ) k ˆ A ) ˆ B ) = 0 for k ≥ ρ, ˆ A ] = 0 and/or [ ˆ ρ, ˆ B ] = 0 In this appendix, we attempt to expand ((ˆ δ ˆ ρ ) k ˆ A ) and to show the following identityexplicitly when [ˆ ρ, ˆ A ] = 0 and/or [ˆ ρ, ˆ B ] = 0 are hold:Tr (cid:16) f (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ˆ A (cid:17) ˆ B (cid:17) = 0 ( k ≥ . (A.1)First, we expand the quantity ((ˆ δ ˆ ρ ) k ˆ A ) by considering the definition of ˆ δ ρ : ˆ δ ˆ ρ ˆ A =ˆ ρ ˆ A − ˆ A ˆ ρ . ((ˆ δ ˆ ρ ) k ˆ A ) = k X j =0 C ( k ) j ˆ ρ j ˆ A ˆ ρ k − j . (A.2)11e attempt to find the recursion formula of C ( k ) j to obtain the expression of C ( k ) j . Byapplying ˆ δ ˆ ρ to (ˆ δ ˆ ρ ) k − ˆ A , we have C ( k ) j = C ( k − j − − C ( k − j (1 ≤ j ≤ k − , (A.3a) C ( k ) k = C ( k − k − , (A.3b) C ( k )0 = − C ( k − . (A.3c)Next, we expand ( x − y ) k as( x − y ) k = k X j =0 D ( k ) j x j y k − j , D ( k ) j = ( − k − j (cid:18) kj (cid:19) . (A.4)By applying ( x − y ) to ( x − y ) k − , we obtain D ( k ) j = D ( k − j − − D ( k − j (1 ≤ j ≤ k − , (A.5a) D ( k ) k = D ( k − k − , (A.5b) D ( k )0 = − D ( k − . (A.5c)The recursion formula of C ( k ) j is equivalent to that of D ( k ) j . The initial conditions aregiven by C (1)0 = − , C (1)1 = 1 , (A.6a) D (1)0 = − , D (1)1 = 1 . (A.6b)Therefore, C ( k ) j equals D ( k ) j . We obtain C ( k ) j = D ( k ) j = ( − k − j (cid:18) kj (cid:19) . (A.7)We have the expansion of ((ˆ δ ˆ ρ ) k ˆ A ), Eq. (A.2) with Eq. (A.7).With the above result, we attempt to prove the equation:Tr (cid:16) f (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ˆ A (cid:17) ˆ B (cid:17) = 0 ( k ≥ . (A.8)When the cyclic permutation property of trace is hold and the commutation relation[ˆ ρ, ˆ A ] = 0 and/or [ˆ ρ, ˆ B ] = 0 are hold, we have the following equation by using Eq. (A.2).Tr (cid:16) f (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ˆ A (cid:17) ˆ B (cid:17) = k X j =0 C ( k ) j Tr (cid:16) f (ˆ ρ )ˆ ρ j ˆ A ˆ ρ k − j ˆ B (cid:17) = k X j =0 C ( k ) j Tr (cid:16) f (ˆ ρ )ˆ ρ k ˆ A ˆ B (cid:17) = Tr (cid:16) f (ˆ ρ )ˆ ρ k ˆ A ˆ B (cid:17) k X j =0 C ( k ) j ( k ≥ . (A.9)12rom Eqs. (A.4) and (A.7), we have k X j =0 C ( k ) j = 0 ( k ≥ (cid:16) f (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ˆ A (cid:17) ˆ B (cid:17) = 0 for [ˆ ρ, ˆ A ] = 0 and / or [ˆ ρ, ˆ B ] = 0 ( k ≥ . (A.10)From the result, Eq. (A.10), we have the following identity by substituting δ ˆ ρ into ˆ A ,substituting ˆ Q [ a ] into ˆ B , and replacing f (ˆ ρ ) with g ( k +1) (ˆ ρ ), when ˆ Q [ a ] has the property,[ˆ ρ, ˆ Q [ a ] ] = 0: Tr (cid:16) g ( k +1) (ˆ ρ ) (cid:16) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:17) ˆ Q [ a ] (cid:17) = 0 ( k ≥ . (A.11)The expansion of ((ˆ δ ˆ ρ ) k ˆ A ), Eq. (A.2), may be useful to calculate quantities. Appendix B. Expression of Tr (cid:0) f ( ˆ ρ )((ˆ δ ˆ ρ ) k ( δ ˆ ρ )) ˆ B (cid:1) in the case of [ ˆ ρ, ˆ B ] = r ( ˆ ρ ) We treat the case of [ˆ ρ, ˆ B ] = ˆ ρ ˆ B − ˆ B ˆ ρ = r (ˆ ρ ). For a function u (ˆ ρ ) and an operatorˆ B , we have the following equation with the quantum analysis:[ u (ˆ ρ ) , ˆ B ] = ˆ δ u (ˆ ρ ) ˆ B = ˆ δ u (ˆ ρ ) ˆ δ ˆ ρ ˆ δ ˆ ρ ˆ B = Z dtu (1) (ˆ ρ − t ˆ δ ˆ ρ )[ˆ ρ, ˆ B ] = Z dtu (1) (ˆ ρ − t ˆ δ ˆ ρ ) r (ˆ ρ ) . (B.1)We have [ u (ˆ ρ ) , ˆ B ] = u (1) (ˆ ρ ) r (ˆ ρ ) for [ˆ ρ, ˆ B ] = r (ˆ ρ ) . (B.2)Therefore, we obtain the following equation with [ˆ ρ k − j , ˆ B ] = ( k − j )ˆ ρ k − j − r (ˆ ρ ) by sub-stituting ˆ ρ k − j into u (ˆ ρ ):Tr (cid:16) f (ˆ ρ )ˆ ρ j ( δ ˆ ρ )ˆ ρ k − j ˆ B (cid:17) = Tr (cid:16) f (ˆ ρ )ˆ ρ k ( δ ˆ ρ ) ˆ B (cid:17) + ( k − j )Tr (cid:16) f (ˆ ρ ) r (ˆ ρ )ˆ ρ k − ( δ ˆ ρ ) (cid:17) . (B.3)We have the following equation by using Eq. (A.2):Tr h f (ˆ ρ ) (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1) ˆ B i = Tr h f (ˆ ρ ) (cid:0) k X j =0 C ( k ) j ˆ ρ j ( δ ˆ ρ )ˆ ρ k − j (cid:1) ˆ B i = k X j =0 C ( k ) j Tr h f (ˆ ρ )ˆ ρ j ( δ ˆ ρ )ˆ ρ k − j ˆ B i , (B.4)where C ( k ) j = ( − k − j k ! j !( k − j )! . Inserting Eq. (B.3) into Eq. (B.4), we haveTr h f (ˆ ρ ) (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1) ˆ B i = Tr (cid:16) f (ˆ ρ )ˆ ρ k ( δ ˆ ρ ) ˆ B (cid:17)" k X j =0 C ( k ) j + Tr (cid:16) f (ˆ ρ ) r (ˆ ρ )ˆ ρ k − ( δ ˆ ρ ) (cid:17)" k − X j =0 C ( k ) j ( k − j ) . (B.5)13he sums of C ( k ) j satisfy k X j =0 C ( k ) j = 0 ( k ≥ , (B.6a) k − X j =0 C ( k ) j ( k − j ) = 0 ( k ≥ . (B.6b)As a result, we haveTr h f (ˆ ρ ) (cid:0) (ˆ δ ˆ ρ ) k ( δ ˆ ρ ) (cid:1) ˆ B i = k ≥ − Tr h f (ˆ ρ ) r (ˆ ρ )( δ ˆ ρ ) i ( k = 1)Tr h f (ˆ ρ )( δ ˆ ρ ) ˆ B i ( k = 0) . 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