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 - 1 -

Quantum Thermodynamics Prague, 7 June 2016 - 2 - 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 of quantum systems can be calculated explicitly. ⃝ DC Brody 2016 Analytic and algebraic methods in physics c

Quantum Thermodynamics Prague, 7 June 2016 - 3 - 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 = CP r − 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. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 4 - 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. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 5 - 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 ] = k B log P ( A ) , (2) where k B is Bolzmann’s constant. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 6 - 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 H j ( x ) for the Hamiltonian function of the j th molecule. The weak interaction condition implies that the probability measure factorises: P (d x ) = P 1 (d x 1 ) P 2 (d x 2 ) · · · P n (d x n ) . (3) Then it follows that the H j ( x ) , j = 1 , 2 , . . . , n , when interpreted as functions on 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:   n S ( n ) ( E ) = 1  1 ∑  . n k B log P H j ≤ E (4) n j =1 Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 7 - 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 n →∞ S ( n ) ( E ) S ( E ) = lim (5) for the specific entropy of the bath can then be used to work out the temperature of the bath: d S ( E ) 1 = T ( E ) . (6) d E 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:   n 1  1 ∑  ≤ inf n log P H j ≤ E β ≥ 0 [ βE + log Z ( β )] , (7) n j =1 Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 8 - where Z ( β ) = E [exp ( − βH )] . (8) Recall that for any sequence of real numbers a n , the superior limit is defined by lim sup n →∞ a n = lim n →∞ sup m ≥ n a m and the inferior limit is defined by lim inf n →∞ a n = lim n →∞ inf m ≥ n a m . 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 a n ≤ b for all n , then lim sup n →∞ a n ≤ b . With these facts in mind we deduce that   n 1  1 ∑  ≤ inf lim sup n log P H j ≤ E β ≥ 0 [ βE + log Z ( β )] . (9) n n →∞ j =1 This concludes the first step in the derivation of the thermodynamic limit. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 9 - 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 > E min it holds that P ( H < ϵ ) > 0 . Proposition 1. If P is H -complete, then H e − βH ] [ E lim = E min (10) E [ e − βH ] β →∞ and for any E ∈ ( E min , ¯ E ] there exists a unique value of β ≥ 0 such that H e − βH ] [ E = E (11) E [ e − βH ] . 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 ( E min , ¯ E ] , namely, the value of β such that equation (11) is satisfied. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 10 - Then we have [ ] inf βE + log Z ( β ) = β ( E ) E + log Z ( β ( E )) , (12) β ≥ 0 from which it follows that   n 1  1 ∑  ≤ β ( E ) E + log Z ( β ( E )) . lim sup n log P H j ≤ E (13) n n →∞ j =1 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:   n 1  1 ∑  ≥ β ( E ) E + log Z ( β ( E )) , lim inf n log P H j ≤ E (14) n n →∞ j =1 Now we are in a position to derive the equation of state of the bath. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 11 - Proposition 2. The thermodynamic limit   n 1  1 ∑ S ( E ) = lim n k B log P H j ≤ E (15)  n n →∞ j =1 exists, and the resulting expression for the specific entropy of the heat bath is S ( E ) = k B β ( E ) E + k B log Z ( β ( E )) , (16) where for each value of E ∈ ( E min , ¯ E ] the associated value of β is determined by H e − β ( E ) H ] [ E = E e − β ( E ) H ] , (17) [ E and Z ( β ( E )) = E [exp ( − β ( E ) H )] . Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 12 - 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 (d x ) = 1 ∑ δ i (d x ) . (18) r i It should be apparent that for r = 2 the partition function is e − βE 1 + e − βE 2 ) Z ( β ) = 1 ( (19) . 2 We find that the expression for the specific energy as a function of β is E ( β ) = E 1 e − βE 1 + E 2 e − βE 2 (20) . e − βE 1 + e − βE 2 We can invert this relation, to give β as a function of E , as follows: 1 log E 2 − E β ( E ) = (21) . E 2 − E 1 E − E 1 Analytic and algebraic methods in physics ⃝ DC Brody 2016 c

Quantum Thermodynamics Prague, 7 June 2016 - 13 - Inserting this expression for β in terms of E back into the partition function, we obtain a formula for a partition function as a function of E , given by ( E − E 1 ( E 2 − E   E 1 E 2 E 2 − E 1 + ) ) Z ( β ( E )) = 1 E 2 − E 1  . (22)  2 E 2 − E 1 E 2 − E 1 Finally, inserting (21) and (22) into the thermodynamic relation (16), we obtain the following expression for the specific entropy of the bath: [ 2 − E − E 1 log E − E 1 − E 2 − E log E 2 − E ] log 1 S ( E ) = k B (23) . E 2 − E 1 E 2 − E 1 E 2 − E 1 E 2 − E 1 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. Analytic and algebraic methods in physics ⃝ DC Brody 2016 c