Quantum Fluctuation Relations and the Arrow of Time Peter Hänggi, Institut für Physik, Universität Augsburg M. Campisi, P. Hänggi, and P. Talkner Colloquium: Quantum fluctuation relations: Foundations and applications Rev. Mod. Phys. 83, 771 791 (2011); Addendum and Erratum: Quantum fluctuation relations: Foundations and applications Rev. Mod. Phys. 83, 1653 (2011). M. Campisi and P. Hänggi Fluctuation, Dissipation and the Arrow of Time Entropy 13, 2024 2035 (2012).
Sekhar Burada Michele Campisi Gert Ingold Eric Lutz Manolo Morillo Peter Talkner Acknowledgments
The famous Laws Equilibrium Principle -- minus first Law An isolated, macroscopic system which is placed in an arbitrary initial state within a finite fixed volume will attain a unique state of equilibrium. Second Law (Clausius) For a non-quasi-static process occurring in a thermally isolated system, the entropy change between two equilibrium states is non-negative. Second Law (Kelvin) No work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source.
MINUS FIRST LAW vs. SECOND LAW -1st Law 2nd Law
Second Law Rudolf Julius Emanuel Clausius (1822 1888) Heat generally cannot spontaneously flow from a material at lower temperature to a material ilat higher h temperature. William Thomson alias Lord Kelvin (1824 1907) No cyclic process exists whose sole effect is to extract heat from a single heat bath at temperature T and convert it entirely to work. δq = T ds (Zürich, 1865)
Entropy S content of transformation Verwandlungswert d S = δq rev T ; δq irrev < δq rev V 2,T 2 I δq Γ rev Γ irrev T 0 V 1,T 1 S(V 2,T 2 ) S(V 1,T 1 ) Z Z S(V 2,T 2 ) S(V 1,T 1 )= Γ irrev Γ rev δq I C C = Γ 1 rev + Γ irrev T S δq t 0 NO! T
What are fluctuation and work theorems about? Small systems: fluctuations may become comparable to average quantities. Can one infer thermal equilibrium properties from fluctuations in nonequilibrium processes?
Equilibrium versus nonequilibrium processes Isothermal quasistatic process: e βh(t 0) /Z(t 0 ) BATH β weak contact SYSTEM H(t 0 ) xx xx xxquasixxstatic e βh(t f ) /Z(t f ) BATH β weak contact SYSTEM H(t f ) Non-equilibrium process: H(t 0 ) H(t f ) F = w = w p tf,t 0 (w) =? pdf of work t 0 t f t
T H = H(z, λ t ) λ 0 eq. ρ 0 (z) = e βh(z,λ 0 ) Z(λ 0 ) λ τ non-eq ρ τ (z) e βh(z,λ τ ) Z(λ τ ) λ : [0, τ] R protocol t λ t
MICROREVERSIBILITY OF TIME DEPENDENT SYSTEMS p z τ z 0 ϕ t,0[z 0; λ] Classical q ε(q, p) = (q, p) εz 0 ϕ τ t,0[εz τ ; λ] εϕ t,0 [z 0 ; λ] = ϕ τ t,0 [εz τ ; λ] εz τ i ψt = Ut,0[λ] i f Quantum Θ i Uτ t,0[ λ]θ f Θ f Θ = Time reversal operator U t,τ [λ] = Θ U τ t,0 [ λ]θ
CLASSICAL WORK p z τ z 0 ϕ t,0 [z 0 ; λ] εz 0 W [z 0 ; λ] = H(z τ, λ τ ) H(z 0, λ 0 ) = ϕ τ t,0 [εz τ ; λ] εz τ q dt λ H(ϕ t,0 [z 0 ; λ], λ t ) t λ t p[w ; λ] = dz 0 ρ 0 (z 0 )δ[w H(z τ, λ τ ) + H(z 0, λ 0 )] Work is a RANDOM quantity
JARZYNSKI EQUALITY C. Jarzynski, PRL 78 2690 (1997) e βw = e β F F = F (λ τ ) F (λ 0 ) = β 1 ln Z(λ τ ) Z(λ 0 ) Z(λ t ) = dz e βh(z,λt) Since exp is convex, then W F Second Law
INCLUSIVE VS. EXCLUSIVE VIEWPOINT C. Jarzynski, C.R. Phys. 8 495 (2007) H(z, λ t ) = H 0 (z) λ t Q(z) W = dtq t λ t = H(z τ, λ τ ) H(z 0, λ 0 ) Not necessarily of the textbook form d(displ.) (force) W 0 = dtλ t Qt = H 0 (z τ ) H 0 (z 0 ) z t = {H(z t, λ t ); z} G.N. Bochkov and Yu. E. Kuzovlev JETP 45, 125 (1977)
Canonical initial state Exclusive Work: p[w 0 ; λ] p[ w 0 ; λ] = eβw 0 Bochkov-Kuzovlev-Crooks e βw 0 = 1 Bochkov-Kuzovlev G.N. Bochkov, Y.E. Kuzovlev, Sov. Phys. JETP 45, 125 (1977); G.N. Bochkov, Y.E. Kuzovlev, Physica A 106, 443 (1981)
INCLUSIVE, EXCLUSIVE and DISSIPATED WORK W diss = W F e βw diss = 1 W, W 0, W diss are DISTINCT stochastic quantities p[x; λ] p 0 [x; λ] p diss [x; λ] They coincide for cyclic protocols λ 0 = λ τ M.Campisi, P. Talkner and P. Hänggi, Phil. Trans. R. Soc. A 369, 291 (2011)
GAUGE FREEDOM H (z, t) = H(z, t) + g(t) W = W + g(τ) g(0) F = F + g(τ) g(0) W, F are gauge dependent: not true physical quantities WARNING: be consistent! use same gauge to calculate W and F e βw = e β F e βw = e β F The fluctuation theorem is gauge invariant! Gauge: irrelevant for dynamics crucial for ENERGY-HAMILTONIAN connection
GAUGE FREE vs. GAUGE DEPENDENT EXPRESSIONS OF CLASSICAL WORK H = H 0 (z) λ t Q(z) + g(t) Inclusive work: W = H (z τ, τ) H (z 0, 0), g-dependent W phys = dtq t λt, g-independent W phys = W g(τ) + g(0) Exclusive work: W 0 = H 0 (z τ, λ τ ) H 0 (z 0, λ 0 ) = dtλ t Q t, g-independent Dissipated work: W diss = H (z τ, τ) H (z 0, 0) F, g-independent
QUANTUM WORK FLUCTUATION RELATIONS H(z, λ t ) H(λ t ) ρ(z, λ t ) ϱ(λ t ) = e βh(λt) Z(λ t ) Z(λ t ) Z(λ t ) = Tre βh(λt) [ ϕ t,0 [z 0 ; λ] U t,0 [λ] = T exp i W [z 0 ; λ]? t 0 ] dsh(λ s )
W[λ] = U t,0 [λ]h(λ t)u t,0 [λ] H(λ 0 ) = H H t (λ t ) H(λ 0 ) = t 0 dt λ t H H t (λ t ) λ t
WORK IS NOT AN OBSERVABLE P. Talkner et al. PRE 75 050102 (2007) Work characterizes PROCESSES, not states! (δw is not exact) Work cannot be represented by a Hermitean operator W W [z 0 ; λ] w = Em λτ E λ 0 n two-measurements E λt n =instantaneous eigenvalue: H(λ t ) ψ λt n = E λt n ψ λt n
Probability of inclusive quantum work recall: H(λ t )ϕ n,ν (λ t ) = e n (λ t )ϕ n,ν (λ t ), P n (t) = ν ϕ n,ν(λ t ) ϕ n,ν (λ t ) 1st measurement of energy: e n (λ 0 ) with prob p n = Tr P n (t 0 )ρ(t 0 ), state after measurement: ρ n = P n (t 0 )ρ(t 0 )P n (t 0 )/p n t 0 t f : ρ n (t f ) = U tf,t 0 ρ n U t f,t 0, i U t,t0 / t = H(λ t )U t,t0, U t0,t 0 = 1 2nd measurement of energy: e m (t f ) with prob. p(m n) = Tr P m (t f )ρ n (t f ) p tf,t 0 (w) = n,m δ(w [e m (λ τ ) e n (λ 0 )])p(m n)p n J. Kurchan, arxiv:cond-mat/0007360
Characteristic function of work G tf,t 0 (u) = dw e iuw p tf,t 0 (w) = e iuem(t f ) e iuen(t0) TrP m (t f )U tf,t 0 ρ n U t + f,t 0 p n m,n = m,n Tre iuh(t f ) P m (t f )U tf,t 0 e ih(t 0) ρ n U + t f,t 0 p n = Tre iuh H(t f ) e iuh(t 0) ρ(t 0 ) e iuh(t f ) e iuh(t 0) t0 H H (t f ) = U t f,t 0 H(t f )U tf,t 0, ρ(t 0 ) = n P n (t 0 )ρ(t 0 )P n (t 0 ), ρ(t 0 ) = ρ(t 0 ) [ρ(t 0 ), H(t 0 )] P. Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, 051131 (2008) P.Talkner, E. Lutz, P. Hänggi, Phys. Rev. E 75, 050102(R) (2007)
Work is not an observable Note: G tf,t 0 (u) is a correlation function. If work was an observable, i.e. if a hermitean operator W existed then the characteristic function would be of the form of an expectation value G W (u) = e iuw = Tre iuw ρ(t 0 ) Hence, work is not an observable. P. Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, 051131 (2008) P.Talkner, E. Lutz, P. Hänggi, Phys. Rev. E 75, 050102(R) (2007)
Choose u = iβ e βw = dw e βw p tf,t 0 (w) = Gt c f,t 0 (iβ) = Tre βh H(t f ) e βh(t0) Z 1 (t 0 )e βh(t 0) = Tre βh(t f ) /Z(t 0 ) = Z(t f )/Z(t 0 ) = e β F quantum Jarzynski equality
u u + iβ and time-reversal Z(t 0 )G c t f,t 0 (u) = Tr U + t f,t 0 e iuh(t f ) U tf,t 0 e i( u+iβ)h(t 0) = Tr e i( u+iβ)h(t f ) e βh(t f ) U + t 0,t f e i( u+iβ)h(t 0) U t0,t f = Z(t f )G c t 0,t f ( u + iβ) p tf,t 0 (w) p t0,t f ( w) = Z(t f ) Z(t 0 ) eβw β( F w) = e Tasaki-Crooks theorem white P. Talkner, and P. Hänggi, J. Phys. A 40, F569 (2007).
Microcanonical initial state ρ(t 0 ) = ω 1 E (t 0)δ(H(t 0 ) E), ω E (t 0 ) = Tr δ(h(t 0 E)) = e S(E,t 0)/k B ω E (t 0 ): Density of states, S(E, t 0 ): Entropy p mc t f,t 0 (E, w) = ω 1 E (t 0)Tr δ(h H (t f ) E w)δ(h(t 0 ) E) pt mc f,t 0 (E, w) pt mc 0,t f (E + w, w) = ω E+w (t f ) = e [S(E+w,t f ) S(E,t 0 )]/k B ω E (t 0 ) P.Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, 051131 (2008)
PROBLEM Projective quantum measurement of: E =TOTAL energy N = TOTAL number of particles in MACROSCOPIC systems!!!
Measurements A projective measurement of an observable A = i a ip A i, P A i P A j = δ i,j P A i : ρ i P A i ρp A i while a force protocol is running leads to a change of the statistics of work but leaves unchanged the Crooks relation and the Jarzyninski equality. M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 105, 140601 (2010); M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. E 83, 041114, (2011).
OPEN QUANTUM SYSTEM: WEAK COUPLING β H S(λ t) H SB H B H(λ t ) = H S (λ t ) + H B + H SB p[ E, Q; λ] = E = Em λτ E λ 0 n ϱ(λ 0 ) = e βh(λ 0) /Y (λ 0 ) system energy change E B = E B µ E B ν = Q bath energy change δ[ E Em λτ + E λ 0 n ]δ[q + Eµ B Eν B ]p mµ nν [λ]pnν 0 m,n,µ,ν p[ E, Q; λ] p[ E, Q; λ] = eβ( E Q F S ) E = w + Q F S = β 1 ln Z S(λ τ ) Z S (λ 0 ) p[w, Q; λ] p[ w, Q; λ] = eβ(w F S ) P. Talkner, M. Campisi, P. Hänggi, J.Stat.Mech. (2009) P02025
Strong coupling: Quantum Treatment H(t) = H S (t) + H SB + H B Fluctuation Theorem for the total system: where: and p tf,t 0 (w) p t0,t f ( w) = Y (t f ) Y (t i ) eβw Y (t) = Tre β(h S (t)+h SB +H B ) w = e m (t f ) e n (t 0 ) e n (t)= instantaneous eigenvalues of total system
OPEN QUANTUM SYSTEM: STRONG COUPLING β H S(λ t) H SB H B H(λ t ) = H S (λ t ) + H SB + H B w = Em λτ E λ 0 n = work on total system=work on S Y (λ t ) = Tre βh(λt) = partition function of total system Z S (λ t ) = Y (λ t) Z B Tr S e βh S (λ t) = partition function of S F S (λ t ) = β 1 ln Z S (λ t ) proper free energy of open system p[w; λ] p[ w; λ] = Y (λ τ ) Y (λ 0 ) eβw = Z(λ τ ) Z(λ 0 ) eβw = e β(w F S ) M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. 102 210401 (2009)
Partition function Z S (t) = Y(t) Z B where Z B = Tr B e βh B
Free energy of a system strongly coupled to an environment Thermodynamic argument: F total system free energy F B bare bath free energy. F S = F F 0 B With this form of free energy the three laws of thermodynamics are fulfilled. G.W. Ford, J.T. Lewis, R.F. O Connell, Phys. Rev. Lett. 55, 2273 (1985); P. Hänggi, G.L. Ingold, P. Talkner, New J. Phys. 10,115008 (2008); G.L. Ingold, P. Hänggi, P. Talkner, Phys. Rev. E 79, 0611505 (2009).
An important difference Route I E. = E S = H S = Tr S+B(H S e βh ) Tr S+B (e βh ) Quantum Brownian motion and the 3 rd law Route II Z = Tr S+B(e βh ) Tr B (e βh B ) U = lnz β Specific heat and dissipation Two approaches Microscopic model Route I U = H H B B ] = E S + [ H SB + H B H B B Route II specific heat density of states Conclusions For finite coupling E and U differ!
Quantum Hamiltonian of Mean Force where also Z S (t) := Y (t) Z B = Tr S e βh (t) H (t) := 1 β ln Tr Be β(h S (t)+h SB +H B ) Tr B e βh B e βh (t) Z S (t) = Tr Be βh(t) Y (t) M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009).
ARROW OF TIME + : movie in forward direction : movie in backward direction frame t : λ t and Q t Forward: βw 0 1 Backward: After Seeing movie βw 0 1 (W 0 is odd under time reversal!) Prior: P(+) = 1/2; P( ) = 1/2 Posterior: P(+ W 0 ) = P(W 0 +)P(+) = P(W 0 ) P(W 0 +)P(+) = P(W 0 +)P(+) + P(W 0 )P( )
USING EXCLUSIVE WORK H(q, p; λ t ) = H 0 (q, p) λ t Q(q, p) λ t = 0 at t = 0; Protocol: λ t : 0 t = τ ϱ 0 (q, p) = e βh 0(q,p) /Z 0 (β) = F.-T. P[Γ, λ] = P[ Γ, λ]e βw 0 B.-K. (1977) W 0 = τ 0 dtλ t Q t λt = λ τ t = e βw 0 λ = 1 equality (2 nd Law) with exp(x) exp( x ) = e βw 0 λ e βw 0 λ = 0 β W 0 λ W 0 λ 0 inequality
ARROW OF TIME F.-T.: P(W 0 +) P(W 0 ) = exp(βw 0) W 0 Odd: P( W 0 ) = P(W 0 +) = P(+ W 0 ) = 1 e βw 0 + 1 1 0.75 P (+ W0) 0.5 0.25 0 10 5 0 5 10 W 0 [k BT ]
ARROW OF TIME 1) H 0 (q, p) H (q, p; t) : emergence of an arrow 2) preparation, here canonical determines its direction conventional : coarsegrainingofquantities< > breakingdet. balance Stoßzahlansatz initiali i molecular l chaos (Bogoliubov) Fokker Planck eqs.; master eqs. All containad hoc elements that induce thetime arrow : future past
Summary Classical and quantum mechanical expressions for the distribution of work for arbitrary initial states The characteristic function of work is given by a correlation function Work is not an observable. Fluctuation theorems (Crooks and Jarzynski) for thermal equilibrium states and microscopic reversibility Open Systems Strong coupling: Fluctuation and work theorems Weak coupling: Fluctuation theorems for energy and heat as well as work and heat Measurements do not affect Crooks relation and Jarzynski equality
References P. Talkner, E. Lutz, P. Hänggi, Phys. Rev. E, 75, 050102(R) (2007). P. Talkner, P. Hänggi, J Phys. A 40, F569 (2007). P. Talkner, P. Hänggi, M. Morillo, Phys. Rev. E 77, 051131 (2008). P. Talkner, P.S. Burada, P. Hänggi, Phys. Rev. E 78, 011115 (2008). P. Hänggi, G.-L. Ingold, P. Talkner, New J. Phys. 10, 115008 (2008). P. Talkner, M. Campisi, P. Hänggi, J. Stat. Mech., P02025 (2009). G.-L. Ingold, P. Talkner, P. Hänggi, Phys. Rev. E 79, 0611505 (2009). M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009). M. Campisi, P. Talkner, P. Hänggi, J. Phys. A 42, F392002 (2009). M. Campisi, D. Zueco, P. Talkner, Chem. Phys. 375, 187 (2010). M. Campisi, P. Talkner, P. Hänggi, Phil. Trans. Roy. Soc. 369, 291 (2011). M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 105, 140601 (2010). M. Campisi, P. Hänggi, P. Talkner, Rev. Mod. Phys. 83, 771 (2011). M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. E, 83, 041114 (2011). J. Yi, P. Talkner, Phys. Rev. E 83, 41119 (2011). J. Yi, P. Talkner, M. Campisi, Phys. Rev. E, 84, 011138 (2011).
A. Classical: w in = w ex = τ Definition of work dt dh(z(t), λ t) = H(z(t f ), λ τ ) H(z(t 0 ), λ 0 ) t 0 dt inclusive dt dh(z(t), λ 0) = H(z(t f ), λ 0 ) H(z(t 0 ), λ 0 ) t 0 dt exclusive τ z(t) z(t, z): Trajectory in phase space: ż = {H(z, λ t ), z} z z(t 0 ): Starting point taken from Z 1 (t 0 )e βh(λ 0) for H(z, λ t ) = H 0 (z) λ t Q(z): τ w in = w ex = t 0 τ t 0 dt Q(z(t)) λ t dt Q(z(t))λ t
INCLUSIVE VS. EXCLUSIVE VIEWPOINT C. Jarzynski, C.R. Phys. 8 495 (2007) H(z, λ t ) = H 0 (z) λ t Q(z) W = H(z τ, λ τ ) H(z 0, λ 0 ) = dtq t λ t W 0 = H 0 (z τ ) H 0 (z 0 ) = dtλ t Q t e βw 0 = 1 G.N. Bochkov and Yu. E. Kuzovlev JETP 45, 125 (1977)
Strong coupling: Example System: Two-level atom; bath : Harmonic oscillator H = ɛ ( 2 σ z + Ω a a + 1 ) ( + χσ z a a + 1 ) 2 2 H = ɛ 2 σ z + γ ɛ = ɛ + χ + 2 ( e βω ) β artanh sinh(βχ) 1 e βω cosh(βχ) γ = 1 ( 1 2e βω 2β ln cosh(βχ) + e 2βΩ ) (1 e βω ) 2 Z S = Tre βh S S = F S T F S = k b T ln Z S C S = T S S T M. Campisi, P. Talkner, P. Hänggi, J. Phys. A: Math. Theor. 42: 392002 (2009)
Entropy and specific heat Ω / ε (a) 2 Ω / ε (a) 1 1.8 0.9 1.6 0.8 χ/ε 1 0 1.4 1.2 1 χ/ε 1 0 0.7 0.6 0.5 Ω/ɛ = 3 0.8 0.4 1 0.6 1 0.3 0.4 0.2 0.2 0.1 Ω / ε 0 0 0.6 1.2 k B T/ε Ω / ε 0 0 0.6 1.2 k B T/ε Ω / ε (b) 2 Ω / ε (b) 2 1.8 1.6 1.5 χ/ε 0 S S =0 1.4 1.2 1 0.8 χ/ε 0 C S =0 1 Ω/ɛ = 1/3 S S <0 0.6 0.4 C S <0 0.5 0.2 0 Ω / ε 0 0.6 1.2 k B T/ε 0 Ω / ε 0 0.6 1.2 M. Campisi, P. Talkner, P. Hänggi, J. Phys. A: Math. Theor. 42: 392002 (2009) k B T/ε
Doing small systems: Fluctuation relations and the arrow of time Peter Hänggi, Institut für Physik, Universität Augsburg M. Campisi, P. Hänggi, and P. Talkner Colloquium: Quantum fluctuation relations: Foundations and applications Rev. Mod. Phys. 83, 771 791 (2011); Addendum and Erratum: Quantum fluctuation relations: Foundations and applications Rev. Mod. Phys. 83, 1653 (2011). M. Campisi and P. Hänggi Fluctuation, Dissipation and the Arrow of Time Entropy 13, 2024 2035 (2012). G. L. Ingold, P. Hänggi, and P. Talkner Specific heat anomalies of open quantum systems Phys. Rev. E 79, 061105 (2009)
Characteristic function of work G tf,t 0 (u) = dw e iuw p tf,t 0 (w) = Tre iuh H(λ τ ) e iuh(λ 0) ρ(t 0 ) e iuh H(λ τ ) e iuh(λ0) t0 H H (λ τ ) = U t f,t 0 H(λ τ )U tf,t 0, ρ(t 0 ) = n P n (t 0 )ρ(t 0 )P n (t 0 ) Note: G tf,t 0 (u) is a correlation function. If work was an observable, i.e. if a hermitean operator W existed then the characteristic function would be of the form of an expectation value G W (u) = e iuw = Tre iuw ρ(t 0 ) Hence, work is not an observable. P. Talkner, P. Hänggi, M. Morillo, PRE 77, 051131 (2008) P.Talkner, E. Lutz, P. Hänggi, Phys. Rev. E 75, 050102(R) (2007)
MORE ON GAUGE FREEDOM W phys = dtq t λ t Corresponds to the choice g = 0. Other physical expressions are possible corresponding to different choices of g. The experimental conditions determine uniquely the SPECIFIC gauge one must choose!!!