Quantum Thermodynamics Dorje C. Brody Department of Mathematics - - PowerPoint PPT Presentation

Quantum Thermodynamics

Dorje C. Brody Department of Mathematics Brunel University London Uxbridge UB8 3PH, United Kingdom — Dorje.Brody@brunel.ac.uk Villa Lanna, Prague, 7 June 2016 Joint work with Lane P. Hughston, Brunel University London

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Quantum Thermodynamics

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Prague, 7 June 2016

• 1. Introduction

We consider the thermodynamics of a quantum heat bath. The heat bath is assumed to consist of a large number of weakly interacting “molecules”, each of which has finitely many degrees of freedom. We assume that the molecular interactions are sufficiently weak. Our goal is to set up the problem in such a way that thermodynamic properties

• f quantum systems can be calculated explicitly.

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Quantum Thermodynamics

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• 2. Quantum state space as probability space

There are three ingredients required for the development of a statistical model for a finite-dimensional quantum system. These are: (i) the phase space of the system, denoted Γ, (ii) the system Hamiltonian ˆ H, and (iii) a normalised measure P on Γ, which determines how “averages” are taken over Γ. We take the system to be modelled by a Hilbert space H of finite dimension r. The phase space of the system is the complex projective space Γ = CPr−1 given by the space of rays through the origin in H. The pair (Γ, F) is then a measurable space, where F denotes the Borel sigma-algebra generated by the open sets of Γ. The use of the term “phase space” in the present context is justified by the fact that Γ has a natural symplectic structure.

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Quantum Thermodynamics

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From ˆ H one can construct an associated Hamiltonian function. This is given for each point x ∈ Γ by the expectation: H(x) = ⟨x| ˆ H|x⟩ ⟨x|x⟩ . (1) The choice of a priori probability measure P on (Γ, F) is not fixed in advance, except that it must be natural to the physical problem under consideration, which for equilibrium typically means either (a) the uniform distribution or (b) a distribution associated with the Hamiltonian. The quantum phase space has the structure of a probability space (Γ, F, P), upon which the Hamiltonian function H : Γ → R is a random variable.

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• 3. Entropy of a subspace of a state space

We shall take the view that the entropy of a quantum system can be expressed as a function of the number of microstates accessible to it: Definition 1. The entropy associated with a measurable subset A ⊂ Γ of a quantum phase space (Γ, F, P) with measure P is given by S[A] = kB log P(A), (2) where kB is Bolzmann’s constant.

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• 4. Construction of quantum heat bath

Now suppose we consider a quantum heat bath consisting of n molecules, all of the same type for simplicity. We write Hj(x) for the Hamiltonian function of the jth molecule. The weak interaction condition implies that the probability measure factorises: P(dx) = P1(dx1) P2(dx2) · · · Pn(dxn). (3) Then it follows that the Hj(x), j = 1, 2, . . . , n, when interpreted as functions

• n the bath state space, are iid random variables under P.

To develop a theory of the thermodynamics we shall take a definition of the specific entropy associated with a given value E of the specific energy: S(n)(E) = 1 n kB log P  1 n

n

∑

j=1

Hj ≤ E   . (4)

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Quantum Thermodynamics

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• 5. Thermodynamic limit

Our strategy will be to show that for fixed E the sequence S(n)(E), n ∈ N, converges for large n. The resulting expression S(E) = lim

n→∞ S(n)(E)

(5) for the specific entropy of the bath can then be used to work out the temperature of the bath: dS(E) dE = 1 T(E). (6) To show that S(n)(E) converges we shall use a variant of Cram´ er’s method in the theory of large deviations. A line of arguments leads to the the following tight bound: 1 n log P  1 n

n

∑

j=1

Hj ≤ E   ≤ inf

β≥0 [βE + log Z(β)] ,

(7)

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Quantum Thermodynamics

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where Z(β) = E [exp (−βH)] . (8) Recall that for any sequence of real numbers an, the superior limit is defined by lim supn→∞ an = limn→∞ supm≥n am and the inferior limit is defined by lim infn→∞ an = limn→∞ infm≥n am. In general the superior limit and the inferior limit need not be the same, but if they agree, then their common value is defined to be the limit of the sequence. One can show that the superior limit has the property that if b is a constant and if an ≤ b for all n, then lim supn→∞ an ≤ b. With these facts in mind we deduce that lim sup

n→∞

1 n log P  1 n

n

∑

j=1

Hj ≤ E   ≤ inf

β≥0 [βE + log Z(β)] .

(9) This concludes the first step in the derivation of the thermodynamic limit.

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Quantum Thermodynamics

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• 6. Completeness condition

We examine in more detail the expression appearing on the right side of (9). Definition 2. We say that the measure P is H-complete if for any ϵ > Emin it holds that P(H < ϵ) > 0. Proposition 1. If P is H-complete, then lim

β→∞

E [ He−βH] E [e−βH] = Emin (10) and for any E ∈ (Emin, ¯ E ] there exists a unique value of β ≥ 0 such that E = E [ He−βH] E [e−βH] . (11) We conclude that if P is H-complete then there exists a unique value of β ≥ 0 such that the infimum is obtained for any given value of E in the range (Emin, ¯ E], namely, the value of β such that equation (11) is satisfied.

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Quantum Thermodynamics

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Then we have inf

β≥0

[ βE + log Z(β) ] = β(E) E + log Z(β(E)), (12) from which it follows that lim sup

n→∞

1 n log P  1 n

n

∑

j=1

Hj ≤ E   ≤ β(E) E + log Z(β(E)). (13)

• 7. Fundamental thermodynamic relation

By the use of the law of large numbers and a change-of-measure technique, a second inequality can be obtained. It is of a similar nature to the inequality above, but runs the other way around: lim inf

n→∞

1 n log P  1 n

n

∑

j=1

Hj ≤ E   ≥ β(E) E + log Z(β(E)), (14) Now we are in a position to derive the equation of state of the bath.

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Proposition 2. The thermodynamic limit S(E) = lim

n→∞

1 n kB log P  1 n

n

∑

j=1

Hj ≤ E   (15) exists, and the resulting expression for the specific entropy of the heat bath is S(E) = kB β(E) E + kB log Z(β(E)), (16) where for each value of E ∈ (Emin, ¯ E] the associated value of β is determined by E = E [ He−β(E)H] E [ e−β(E)H] , (17) and Z(β(E)) = E [exp (−β(E)H)].

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• 8. Example 1: Dirac measure

To gain further intuition about the thermodynamics of a quantum heat bath, it will be instructive if we examine some specific examples. We begin with the Dirac measure: P(dx) = 1 r ∑

i

δi(dx). (18) It should be apparent that for r = 2 the partition function is Z(β) = 1

2

( e−βE1 + e−βE2) . (19) We find that the expression for the specific energy as a function of β is E(β) = E1 e−βE1 + E2 e−βE2 e−βE1 + e−βE2 . (20) We can invert this relation, to give β as a function of E, as follows: β(E) = 1 E2 − E1 log E2 − E E − E1 . (21)

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Inserting this expression for β in terms of E back into the partition function, we

• btain a formula for a partition function as a function of E, given by

Z(β(E)) = 1 2   ( E − E1 E2 − E1 )

E1 E2−E1 +

( E2 − E E2 − E1 )

E2 E2−E1

  . (22) Finally, inserting (21) and (22) into the thermodynamic relation (16), we obtain the following expression for the specific entropy of the bath: S(E) = kB [ log 1

2 − E − E1

E2 − E1 log E − E1 E2 − E1 − E2 − E E2 − E1 log E2 − E E2 − E1 ] . (23) One should bear in mind that all the formulae above are to be interpreted as being applicable at the macroscopic level. We see that the specification of the Hamiltonian of a representative molecule at the microscopic level, along with the specification of the relevant measure on the state space of the molecule (in this case, the Dirac measure), is sufficient to determine completely the equation of state of the bath, in the form of the relation between the energy and the entropy of the bath system as a whole.

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• 9. Example 2: uniform measure

Now we turn to the uniform measure, or Haar measure, which in the case of a single molecule is given by an expression of the form P(dx) = 1 VΓ dVx . (24) Here dVx denotes the natural volume element associated with the Fubini-Study metric on Γ, and VΓ is the total volume of Γ. In the case r = 2 we find that Z(β) = E [ e−βH] = 1 β(E2 − E1) ( e−βE1 − e−βE2) . (25) As a consequence we find that the energy is given as a function of β by E(β) = 1 β + E1 e−βE1 − E2 e−βE2 e−βE1 − e−βE2 . (26)

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• 10. Concluding remarks

In summary, we have introduced an exact model for the thermodynamics of a quantum heat bath. The model depends on the choice of the Hamiltonian for the constituents of the bath, and also on the choice of measure on the phase space of a representative constituent. With these ingredients in place, we have shown that the thermodynamic limit exists, and gives rise to an equation of state that determines a canonical relation between the specific entropy and the specific energy of the bath. Thus from the structure of the molecular state space at a microscopic level we are able to infer the macroscopic properties of the bath. For further details, see D. C. Brody & L. P. Hughston, “Thermodynamics of Quantum Heat Bath” (arXiv:1406.5780).

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