Meaning of temperature in different thermostatistical ensembles - PowerPoint PPT Presentation

Meaning of temperature in different thermostatistical ensembles Peter Hänggi Universität Augsburg In collaboration with: Stefan Hilbert MPI Munich Jörn Dunkel MIT Boston

The famous Laws Equilibrium Principle -- minus first Law An isolated, macroscopic An isolated, macroscopic system ystem which hich is is placed placed in an arbitrary in an arbitrary initial initial state state within ithin a finite fixed finite fixed volume olume will attain ill attain a uni a unique que state state of equilibrium. f equilibrium. Second Law (Clausius) For a non- For a non-quasi-static quasi-static process process occurring ccurring in a thermally in a thermally isolated isolated system, the system, the entropy ntropy change change between between two two equilibrium quilibrium states states is is non- non-negative. negative. Second Law (Kelvin) No work can No work can be extracted xtracted from rom a closed closed equilibrium equilibrium system system during during a cyclic cyclic variation variation of a parameter f a parameter by an external n external source. source.

SECOND LAW Quote by Sir Arthur Stanley Eddington: “If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.“ Freely translated into German: Falls Ihnen jemand zeigt, dass Ihre Lieblingstheorie des Universums nicht mit den Maxwellgleichungen übereinstimmt - Pech für die Maxwellgleichungen. Falls die Beobachtungen ihr widersprechen - nun ja, diese Experimentatoren bauen manchmal Mist. -- Aber wenn Ihre Theorie nicht mit dem zweiten Hauptsatz der Thermodynamik übereinstimmt, dann kann ich Ihnen keine Hoffnung machen; ihr bleibt nichts übrig als in tiefster Schande aufzugeben.

MINUS FIRST LAW vs. SECOND LAW -1st Law 2nd Law

Thermodynamic Temperature e ody a e pe a u e δ Q rev = T dS ← thermodynamic entropy S = S ( E, V, N 1 , N 2 , ... ; M, P, ... ) S ( E, ... ): (continuous) & di ff erentiable and S ( E ) ( ti ) & di ff ti bl d monotonic function of the internal energy E µ ∂ S µ ¶ ¶ = 1 ∂ E ∂ E T T ...

micr croc ocanoni anonical ensem nsembl ble

Entropy in Stat. Mech. S = k B ln ­ ( E; V; ::: ) X QM: ­ G ( E; V; ::: ) = 1 0 · E i · E classical µ ¶ Z ¡ ¢ 1 E ¡ H ( q; p ; V; ::: ) Gibbs: ­ G = d¡ £ DOF N ! h Z ¡ ¢ @ ­ G / E ¡ H ( q; p ; V; ::: ) Boltzmann: ­ B = ² 0 d¡ ± @E density of states

Entropy in Stat. Mech. S = k B ln ­ ( E; V; ::: ) µ ¶ Z ¡ ¢ 1 E ¡ H ( q; p ; V; ::: ) Gibbs: ­ G = d¡ £ DOF N ! h Z ¡ ¢ @ ­ G / E ¡ H ( q; p ; V; ::: ) Boltzmann: ­ B = ² 0 d¡ ± @E density of states

Microcanonical thermostatistics Ω ω D-Operator DoS ! ( E, Z ) = Tr[ � ( E � H )] � 0 ⇢ ( ξ | E, Z ) = � ( E � H ) ! Ω( E, Z ) = Tr[Θ( E � H )] IntDoS Thermodynamic Entropy ? S B ( E ) = ln ( ✏ ! ) S G ( E ) = ln Ω . vs. Boltzmann (?) Gibbs (1902), Hertz (1910)

Boltzmann vs . Gibbs Ω ω S B ( E ) = ln ( ✏ ! ) S G ( E ) = ln Ω . ✓ @S ◆ − 1 T ( E, Z ) ⌘ @E T G ( E ) = Ω T B ( E ) = ! ! ⌫ ⌫ ( E, Z ) = @!/@E,

Density of states of the pendulum in reduced units (complete elliptic integrals of the first kind). Fig. 1 in reference: M. Baeten and J. Naudts, Entropy, 13, 1186 ‐ 1199 (2011).

N Spins ǀ S ǀ = 1/2 Entropy for N = 100 N = 100 (magenta: S _G ; blue: S _B N = 10 8 Δ = M B – M = ‐ k B T B /B

g SCIENCE VOL 339 4 JANUARY 2013 Negative Absolute Temperature for Motional Degrees of Freedom S. Braun, 1,2 J. P. Ronzheimer, 1,2 M. Schreiber, 1,2 S. S. Hodgman, 1,2 T. Rom, 1,2 I. Bloch, 1,2 U. Schneider 1,2 * Because negative temperature systems can ab- sorb entropy while releasing energy, they give rise to several counterintuitive effects, such as Carnot engines with an efficiency greater than ✓ Carnot efficiencies >1 unity ( 4 ). Through a stability analysis for thermo- dynamic equilibrium, we showed that negative temperature states of motional degrees of free- ✓ Dark Energy dom necessarily possess negative pressure ( 9 ) and are thus of fundamental interest to the description of dark energy in cosmology, where negative pres- sure is required to account for the accelerating expansion of the universe ( 10 ). Cold atoms in optical lattices are an ideal

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‘Non-uniqueness’ of temperature  E 2 ✏ − 1 ✓ 2 E ◆� + E Ω ( E ) = exp 4 sin 2 ✏ , ✏ Temperature does NOT determine direction heat flow. Energy is primary control parameter of MCE.

Second Law ! after S i ≥ ∑ j before S j ∑ i

Second law Gibbs S G ( E ) = ln Ω . Ω ( E A + E B ) Z E A + E B d E 0 Ω A ( E 0 ) ! B ( E A + E B � E 0 ) = 0 Z E A + E B Z E 0 d E 0 d E 00 ! A ( E 00 ) ! B ( E A + E B � E 0 ) = 0 0 Z E A + E B Z E A d E 0 d E 00 ! A ( E 00 ) ! B ( E A + E B � E 0 ) � 0 E A Z E A Z E B d E 00 ! A ( E 00 ) d E 000 ! B ( E 000 ) = 0 0 = Ω A ( E A ) Ω B ( E B ) . S G AB ( E A + E B ) ≥ S G A ( E A ) + S G B ( E B ) .

Second Law H A = E A H B = E B before A B coupling S A ( E A ) S B ( E B ) H AB = H A + H B = E A + E B = E AB after A B coupling ! S AB ( E AB ) ≥ S A ( E A ) + S B ( E B )

Second law Boltzmann S B ( E ) = ln ( ✏ ! ) Z E A + E B d E 0 ! A ( E 0 ) ! B ( E A + E B − E 0 ) ✏! ( E A + E B ) = ✏ 0 an ✏ 2 ! A ( E A ) ! B ( E B ). ≥ tween Boltzmann en-

Erunt multi qui, postquam mea scripta legerint, non ad contemplandum utrum vera sint quae dixerim, mentem convertent, sed solum ad disquirendum quomodo, vel iure vel iniuria, rationes meas labefactare possent. G. Galilei, Opere (Ed. Naz., vol. I, p. 412) There will be many who, when they will have read my paper, will apply their mind, not to examining whether what I have said is true, but only to seeking how, by hook or by crook, they could demolish my arguments.

First law X dE = � Q + � A = T dS � p n dZ n , n ✓ @ S ⌧ @ H ◆ � ! p j = T = � , @ Z j @ Z j E,Z n 6 = Z j E Gibbs Boltzmann see also Campisi, Physica A 2007

Entropy S ( E ) second law first law zeroth law equip artition Eq. (38) Eq. (37) Eq. (20) equipartition Gibbs ln Ω yes yes yes yes � � Penrose ln Ω + ln Ω ∞ � Ω � ln Ω ∞ yes yes no no ⇥ ⇤ Complementary Gibbs ln Ω ∞ � Ω yes yes no no ⇥ ⇤ Di ff erential Boltzmann ln Ω ( E + ✏ ) � Ω ( E ) yes no no no � � Boltzmann ln no no no no ✏! Hilbert, Hänggi & Dunkel, in preparation, 2014

Example 1: Classical ideal gas (2 π m ) dN/ 2 Ω ( E, V ) = α E dN/ 2 V N , α = N ! h d Γ ( dN/ 2 + 1) , S B ( E, V, A ) = k B ln[ ǫω ( E )] , S G ( E, V, A ) = k B ln[ Ω ( E )] , � vs. E = dN � dN � E = 2 − 1 k B T B , 2 k B T G . dN

Example 1: Classical ideal gas (2 π m ) dN/ 2 Ω ( E, V ) = α E dN/ 2 V N , α = N ! h d Γ ( dN/ 2 + 1) , S B ( E, V, A ) = k B ln[ ǫω ( E )] , S G ( E, V, A ) = k B ln[ Ω ( E )] , � vs. E = dN � dN � E = 2 − 1 k B T B , 2 k B T G . dN

canoni nonical ensem nsembl ble

w(E) = - E�) 8(E�- Ha)8(E = 1/: d EA TrA[8(EA - HA)] 1/: dE� Tr8[8(E� - Ha)]8(E - EA = Tr 1/: d E'.BwA(EA) 1/: = fpop d EA fpop dE� WA(E'.B)w8(E�)8(E- EA- E�) =fpE dE� - HA) 1/: 8(EA 8(EA [8(E - H)] = TrA {Tra [8(E - HA - Ha)]} = TrA { Tra [1/: d EA - HA) 1/: EA dE� 8(E� - Ha)8(E - HA - Ha) J} = TrA { Tra [1/: d - EA- E�) J} dE� WB(E�)8(E- EA- E�) dEAwA(E'.B)wa(E- E'.B).

Finite bath coupling Thermal Casimir forces and quantum dissipation Introduction γ Quantum H S H S H SB H B dissipation Thermal Casimir effect Conclusions T The definition of thermodynamic quantities for systems coupled to a bath with finite coupling strength is not unique. P . Hänggi, GLI, Acta Phys. Pol. B 37 , 1537 (2006)

An important difference Quantum = E S = 〈 H S 〉 = Tr S+B ( H S e − β H ) Brownian E . motion and Route I the 3rd law Tr S+B (e − β H ) Specific heat and Z = Tr S+B (e − β H ) dissipation U = − ∂ ln Z Route II Two approaches Tr B (e − β H B ) Microscopic model ∂β Route I Route II specific heat ⇒ U = 〈 H 〉−〈 H B 〉 B density of states Conclusions � � = E S + 〈 H SB 〉+ 〈 H B 〉−〈 H B 〉 B For finite coupling E and U differ!

Strong coupling: Example System: Two-level atom; “bath”: Harmonic oscillator Fluctuation Theorem for Arbitrary H = ✏ ✓ a † a + 1 ◆ ✓ a † a + 1 ◆ Open 2 � z + Ω + �� z Quantum 2 2 Systems H ∗ = ✏ ∗ Michele 2 � z + � Campisi e − β Ω sinh( �� ) ✓ ◆ ✏ ∗ = ✏ + � + 2 � artanh 1 − e − β Ω cosh( �� ) ✓ 1 − 2 e − β Ω cosh( �� ) + e − 2 β Ω � = 1 ◆ 2 � ln (1 − e − β Ω ) 2 Z S = Tr e − β H ∗ F S = − k b T ln Z S S S = − @ F S C S = T @ S S @ T @ T M. Campisi, P. Talkner, P. H¨ anggi, J. Phys. A: Math. Theor. 42 392002 (2009)