Energy Fluctuations in Thermally Isolated Driven System

Transcription

1 nergy Fluctuations in Thermally Isolated Driven System Yariv Kafri (Technion) with Guy Bunin (Technion), Luca D Alessio (Boston University) and Anatoli Polkovnikov (Boston University) arxiv: Nature Physics, in press

2 Setup Many body isolated system in a potential Motivation: cold atom systems, trapped ions... t + δt Due to noise in the system the potential is fluctuating in time 2nd law - Lord Kelvin: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work System W>0 fluctuating potential can only increase the energy of the system

3 repeat experiment many times each time random result but we are guaranteed that the average energy increases (and not due to a rare event) t Measured in experiments: cold atom Bose-Hubbard system (3d) H = t ( ) b i b j + b j b i + U n i (n i 1)/2 µ i,j i i n i xtracted from energy measurements and simulations ntropy per Particle (kbn) r 9r 10r 11r Lattice depth (r) HoldTime (ms) ds/dt (kbns-1) Matthias Troyer (TH Zurich)

4 βb = 2A B = 2A + O(N 1 ). ssentially identical question: move piston with a given protocol FIG. 1. Schematic comparison adiabatic limit energy constantbetween - irreversiblethe usual thermal process will increases on average the energy energy increase due to heating (traditional oven, top) and an not-adiabatic work (microwave oven, bottom). On the right of each case we present a schematic picture of the resulting t energy distribution in each case. (i) small work per cycle and (ii) absence of correlations between cycles (see detailed discussion in Methods) the variance of the energy distribution σ 2 () at energy assumes a particularly simple form to leading order in 1/N, where N is the number of degrees of freedom in the system. It depends only on the microcanonical temperature β() = ln Ω() where Ω() is the density of states, and on the average energy change in a cycle at energy, A(): 2 σ () = σ02 A2 () + 2A2 () A2 (0 )! 0 d!. (1) A2 (! ) β (! ) Here 0 is the initial energy of the system and σ02 is the initial variance. This equation is the main result of the paper. The result (1) follows from integrating a Fokker-Planck equation which describs the time evolution of the energy distribution P (, t) (see Methods and Supplementary Material for details): 1 t P = (A()P ) + (B()P ). (2) For interacting systems with many degrees of freed second term on the RHS of this equation is a 1/N tion which can be neglected. This term can be imp though in mesoscopic or integrable systems. q. ( previously suggested for classical systems in Re The fluctuation-dissipation relation (3) is derived a small work assumption (explicitly β()2 "w3 #c $ where "w3 #c is the third cumulant of the wor A() $ T Cv, see Methods and Supplementary rial). As we will show below q. (3) holds for a ver class of classical and quantum systems starting fro interacting particles in a time dependent cavity t interacting spin systems. The main result of the q. (1) is a direct consequence of this relation (see ods). Several interesting consequences follow from (i) When A() is constant the energy width d only on β(), and not on the amplitude of the d other details of the driving protocol. (ii) When A not constant, depending on the functional form o and β(), the variance of the distribution can be and surprisingly, even smaller than the width of th librium Gibbs distribution at the same mean ener 2 fact, σ 2 ()/σeq () can be made arbitrarily sma proper choice of A(). (iii) When A is a func the energy density u = /N (with a possible ex energy independent prefactor like the total num particles), we have σ 2 () O(N ), scaling as in e rium. For a single quench this result was noticed Ref. [14]. Here we show that it remains valid after quenches. (iv) The dependence of σ 2 on displa qualitatively distinct behaviors with increasing pending on whether the integral in q. (1) diver converges as. To illustrate the distinct behaviors associated point (iv) above we consider the generic case β α, which is the case for phonons, superflu

5 Several questions (begin to address in this talk): Can we say something about the distribution of the final energies? (mesoscopic) Given Can we say something about the variance? σ 2 () t can it decrease? t How do they compare to changing the energy of the system by coupling the system to a thermal bath? σ 2 eq() =T 2 C v Can it be wider/narrower? independent of history, given energy know width s s Can one classify different systems with distinct behaviors?

6 Main Result If the drive is slow enough (still irreversible + exact conditions later) the variance is governed by the rate of energy change in the system σ 2 () implies σ 2 () =σ 2 0 t Specifically, given A() = t (depends on how potential varies and for a given system can be controlled to a large extent) we can write: A 2 () A 2 ( 0 ) +2A2 () 0 d A 2 ( ) β ( ) t at initial state inverse temperature at energy In essence a direct result of time-reversal symmetry

7 Outline for the rest of the talk First discuss the implications of the results - in particular observe two broad classes of distributions Illustrate the idea behind derivation in a trivial example Derivation and conditions for the relation to hold Illustrate on another example (driven XY model in1d - we also looked at a driven quantum transverse field Ising model in 1d and particle in a chaotic cavity) Comments - relation to fluctuation relations (which must be obeyed) and quantum mechanics Summarize

8 Implications of results 1 Recall, given A() = t we can write: thermal bath σ 2 eq() =T 2 C v σ 2 () =σ 2 0 A 2 () A 2 ( 0 ) +2A2 () 0 d A 2 ( ) β ( ) at initial state inverse temperature at energy β() If A() is constant the width of the distribution depends only on σ 2 () =σ d β( )

9 Implications of results 2 Recall, given A() = t A we can write: thermal bath σ 2 eq() =T 2 C v σ 2 () =σ 2 0 A 2 () A 2 ( 0 ) +2A2 () 0 d A 2 ( ) β ( ) at initial state inverse temperature at energy Depending on the functional form of A(), β() the distribution can be larger and somewhat surprisingly smaller than the equilibrium distribution.

10 Implications of results - 3 Recall, given A() = t we can write: thermal bath σ 2 eq() =T 2 C v σ 2 () =σ 2 0 A 2 () A 2 ( 0 ) +2A2 () 0 d A 2 ( ) β ( ) at initial state inverse temperature at energy Two distinct behaviors depending on if integral is controlled by upper or lower bound (of course doesn t have to be). Namely, if integral diverges/converges asymptotically at large

11 Illustrate last point for genetic β() α Ideal gas...) (Goldstone modes, Fermi liquid, from positivity of specific heat and demanding an unbounded functions of energy 0 < α 1 Take A() = t = c s, namely, rate of change power law in energy. demanding finite energy at finite time s 1

12 Results normalized by equilibrium width: β() α η =2s 1 α A() = t A Regime Condition Time Scale width Gibbs-like η < 0 1 s s c exp[ ] σ 2 (1 s) η σeq 2 run-away η > 0 1 s s c exp[ ] σ 2 2α (1 s) η σeq 2 η critical η =0 - time needed to reach asymptotic distribution 2α η η 0 Broad classification remains valid as long as functions are monotonic, namely A() σ 2 σ 2 eq 2α log - For large negative η the distribution becomes very narrow S t In terms of entropy (Fokker-Planck for entropy distribution) integral becomes S ds boarder line when S 0 Â 2 (S )

13 Derivation

14 Idea through a (really) simple example Weakly interacting harmonic oscillators F (x) t impulse on one oscillator weak interactions allow to thermalize p p + F (x) t particle momentum Assume impulse short enough that position doesn t change Let system equilibrate between pulses (quasi-static)

15 Using fact that between impulses system equilibrates ρ(x, v) e β() = p 2 + kx2 2 average over initial positions in eq. to obtain the first and second cumulants of the work (p + F (x) t) 2 p 2 note that βb =2A Fluctuation - dissipation relation Comment - results will not change if act on several oscilators

16 Since we are essentially dealing with a quasi-static process we can describe the evolution of the energy by a Fokker-Planck equation t P = (A()P )+ 1 2 (B()P ) but with βb =2A and time the number of impulses Write t = A() t σ 2 = B +2( A() A() ) or σ 2 = B +2( A A ) A

17 Assume narrow probability distribution σ 2 =2β 1 ( )+2 A( ) A( ) σ2 ( ) σ 2 () =σ 2 0 A 2 () A 2 ( 0 ) +2A2 () 0 d A 2 ( ) β ( )

18 Weakly interacting harmonic oscillators weak interactions allow to thermalize For impulse can show Where A() αr β() α Note, here easy to change A() In 1d α =1/2

19 General derivation via Crooks equality (ref: everyone) (will worry about 1/N corrections) Recall - 1. Liouville s theorem quantum mechanically - unitarity (volumes in phase space are conserved under dynamics) 2. Hamiltonian - H(λ(t)) 3. For a given λ dynamics have time reversal symmetry Consider changing λ(t) on isolated system. 0 <t<τ Forward direction - λ(t) Backward direction - λ(τ t) S

20 time reversed λ(t) Isolated system - phase space (microcanonical) after protocol (work done) P F (W, ) = σ Σ P R ( W, + W )= σ Σ +W For periodic driving P F (W, ) P R ( W, + W ) = Σ +W Σ =exp(s( + W ) S()) asy to obtain the same for quantum taking equilibrium density matrix and unitary evolution

21 Difference between reversible and irreversible non-reversible reversible (closed system đq=0)

22 For periodic driving P F (W, ) P R ( W, + W ) = Σ +W Σ =exp(s( + W ) S()) Using S( + W ) S() βw 1 2σeq 2 W 2 P R ( W, + W )=P R ( W, )+W P R ( W, ) Not surprising that we get the Jarzynski relation e βw =1+O( 1 N )

23 To get the Fokker-Planck equation look at cumulant of the work from lne βw βw 2 c =2W + O(1/N ) βb =2A + O(1/N ) t P = (A()P )+ 1 2 (B()P ) For Fokker-Planck to be valid need to demand third cumulant small β 2 w 3 () c w() c This is the quasi-static demand

24 Sloppy version Since we are essentially dealing with a quasi-static process we can describe the evolution of the energy by a Fokker-Planck equation t P = (A()P )+ 1 2 (B()P ) First cumumlant of work at energy w() c Second cumumlant of work at energy w 2 () c Using Liouville or Hamiltonian dynamics one can show that the a stationary solution is the many body density of states P s () =CΩ() With boundary condition P s () =0 < gs this implies J s = A()P s ()+ 1 2 (B()P s ()) = 0 Argument already found in C. Jarzynski 93, D. Cohen 99, Ott 79

25 Results from another example driven XY model in 1d

26 1d XY model rotate spins by small (random) angle 1! 2 /! eq /N

27 Comment We are using fluctuation theorems to derive a fluctuation-dissipation relation. In limit where this fails we can not say anything without more physical information. Suppose that you know mean, you can ask how does this restrict the variance through the fluctuation theorem Variance lower bound Merhav, Kafri for entropy J. Stat. Mech Mean m

28 Trivial to understand P (W ) W W P F (W, ) P R ( W, + W ) = Σ +W Σ =exp(s( + W ) S())

29 Comment In the classical example we looked at the perturbation was sudden. Quantum mechanical this is dangerous! Particle in a well - wave function outside decaying exponential Sudden approximation, say to removal of barrier Ψ before Ψ after 2 Ψ before Ψ after e ikx κx dx

30 Ψ before Ψ after e ikx κx dx This is a power law - due to uncertainty a sudden approximation can push you with non-negligible probability to a high-energy Need a finite time perturbation. Calculations harder - but we could do it for a simple transverse field Ising model H = i gσ x j + σ z j σ z j+1 linearize and modulate in time around g 1 1

31 Summary For driven isolated systems (noisy potential, driving on purpose...) σ 2 () for slow enough driving t t Simple expression - history dependent (in contrast to heating with thermal bath) Broadly two different regimes - equilibrium like and wide run away Can show hold for other examples (XY model in 1d, TF Ising model (quantum)) AND

32 fluctuation-dissipation relations between drift coe (mobility) and diffusion in open systems [13]. In ular, we find βb = 2A B = 2A + O(N 1 ). FIG. 1. Schematic comparison between the usual thermal heating (traditional oven, top) and an energy increase due to not-adiabatic work (microwave oven, bottom). On the right of each case we present a schematic picture of the resulting energy distribution in each case. (i) small work per cycle and (ii) absence of correlations between cycles (see detailed discussion in Methods) the variance of the energy distribution σ 2 () at energy assumes a particularly simple form to leading order in 1/N, where N is the number of degrees of freedom in the system. It depends only on the microcanonical temperature β() = ln Ω() where Ω() is the density of states, and on the average energy change in a cycle at energy, A(): 2 σ () = σ02 A2 () + 2A2 () 2 A (0 )! 0 d!. (1) A2 (! ) β (! ) Here 0 is the initial energy of the system and σ02 is the initial variance. This equation is the main result of the paper. The result (1) follows from integrating a Fokker-Planck equation which describs the time evolution of the energy distribution P (, t) (see Methods and Supplemen- For interacting systems with many degrees of freedo second term on the RHS of this equation is a 1/N c tion which can be neglected. This term can be imp though in mesoscopic or integrable systems. q. ( previously suggested for classical systems in Re The fluctuation-dissipation relation (3) is derived a small work assumption (explicitly β()2 "w3 #c $ where "w3 #c is the third cumulant of the work A() $ T Cv, see Methods and Supplementary rial). As we will show below q. (3) holds for a ver class of classical and quantum systems starting from interacting particles in a time dependent cavity t interacting spin systems. The main result of the q. (1) is a direct consequence of this relation (see ods). Several interesting consequences follow from (i) When A() is constant the energy width de only on β(), and not on the amplitude of the d other details of the driving protocol. (ii) When A not constant, depending on the functional form of and β(), the variance of the distribution can be and surprisingly, even smaller than the width of th librium Gibbs distribution at the same mean ener 2 fact, σ 2 ()/σeq () can be made arbitrarily smal proper choice of A(). (iii) When A is a funct the energy density u = /N (with a possible ext energy independent prefactor like the total num particles), we have σ 2 () O(N ), scaling as in e rium. For a single quench this result was noticed Ref. [14]. Here we show that it remains valid after quenches. (iv) The dependence of σ 2 on displa qualitatively distinct behaviors with increasing pending on whether the integral in q. (1) diver converges as.