Meaning of temperature in different thermostatistical ensembles - - PowerPoint PPT Presentation

Meaning of temperature in different thermostatistical ensembles

Peter Hänggi Universität Augsburg

Stefan Hilbert MPI Munich In collaboration with: 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

• lume 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 No work can 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

• dy a

e pe a u e

δQrev = T dS ← thermodynamic entropy

S = S(E, V, N1, N2, ...; M, P, ...)

S(E ) ( ti ) & diff ti bl d S(E, ...): (continuous) & differentiable and monotonic function of the internal energy E

µ ∂S ∂E ¶ = 1 T µ ∂E ¶

...

T

micr croc

• canoni

anonical ensem nsembl ble

Entropy in Stat. Mech.

S = kB ln ­(E; V; :::)

Gibbs: ­G = µ 1 N! h

DOF

¶ Z d¡ £ ¡ E ¡ H(q; p; V; :::) ¢ Boltzmann: ­B = ²0 @ ­G @E / Z d¡ ± ¡ E ¡ H(q; p; V; :::) ¢ density of states

QM: ­G(E; V; :::) = X

0·Ei·E

1

classical

Entropy in Stat. Mech.

S = kB ln ­(E; V; :::)

Gibbs: ­G = µ 1 N! h

DOF

¶ Z d¡ £ ¡ E ¡ H(q; p; V; :::) ¢ Boltzmann: ­B = ²0 @ ­G @E / Z d¡ ± ¡ E ¡ H(q; p; V; :::) ¢ density of states

DoS IntDoS D-Operator Boltzmann (?) Gibbs (1902), Hertz (1910)

vs.

Microcanonical thermostatistics

SB(E) = ln (✏ !)

Ω(E, Z) = Tr[Θ(E H)]

!(E, Z) = Tr[(E H)] 0

⇢(ξ|E, Z) = (E H) !

Thermodynamic Entropy ? ω Ω SG(E) = ln Ω.

Boltzmann vs. Gibbs

SB(E) = ln (✏ !)

T(E, Z) ⌘ ✓ @S @E ◆−1

⌫(E, Z) = @!/@E,

TB(E) = ! ⌫

TG(E) = Ω !

ω Ω SG(E) = ln Ω.

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 ǀSǀ = 1/2

Entropy for N = 100 (magenta: S_G ; blue: S_B N = 100 N = 108 Δ = MB – M = ‐ kBTB/B

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. Schneider1,2*

g SCIENCE VOL 339 4 JANUARY 2013 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 unity (4). Through a stability analysis for thermo- dynamic equilibrium, we showed that negative temperature states of motional degrees of free- dom necessarily possess negative pressure (9) and are thus of fundamental interest to the description

• f 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

✓ Carnot efficiencies >1 ✓ Dark Energy

** 23 pages **

‘Non-uniqueness’ of temperature

Temperature does NOT determine direction heat flow. Energy is primary control parameter of MCE.

Ω(E) = exp  E 2✏ − 1 4 sin ✓2E ✏ ◆ + E 2✏,

Second Law

∑i

after Si ≥ ∑j before S j

!

Ω(EA + EB) = Z EA+EB dE0 ΩA(E0)!B(EA + EB E0) = Z EA+EB dE0 Z E0 dE00!A(E00)!B(EA + EB E0)

• Z EA+EB

EA

dE0 Z EA dE00!A(E00)!B(EA + EB E0) = Z EA dE00!A(E00) Z EB dE000!B(E000) = ΩA(EA) ΩB(EB).

SGAB(EA + EB) ≥ SGA(EA) + SGB(EB).

Gibbs

Second law

SG(E) = ln Ω.

Second Law

A B SA(EA) SB(EB) before coupling HA = EA HB = EB A B SAB(EAB) ≥ SA(EA) + SB(EB) after coupling HAB = HA + HB = EA + EB = EAB !

Boltzmann

Second law

SB(E) = ln (✏ !)

✏!(EA +EB) = ✏ Z EA+EB dE0!A(E0)!B(EA +EB −E0)

an ✏2!A(EA)!B(EB). 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

Gibbs

see also Campisi, Physica A 2007

Boltzmann

pj = T ✓ @S @Zj ◆

E,Zn6=Zj !

= ⌧ @H @Zj

• E

,

dE = Q + A = T dS

X

n

pndZn,

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 Differential 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

E = dN 2 − 1

• kBTB,

dN

• E = dN

2 kBTG.

SB(E, V, A) = kB ln[ǫω(E)],

SG(E, V, A) = kB ln[Ω(E)],

vs.

Ω(E, V ) = αEdN/2V N, α = (2πm)dN/2 N!hdΓ(dN/2 + 1),

Example 1: Classical ideal gas

E = dN 2 − 1

• kBTB,

dN

• E = dN

2 kBTG.

SB(E, V, A) = kB ln[ǫω(E)],

SG(E, V, A) = kB ln[Ω(E)],

vs.

Ω(E, V ) = αEdN/2V N, α = (2πm)dN/2 N!hdΓ(dN/2 + 1),

canoni nonical ensem nsembl ble

w(E) = Tr [8(E

• H)]

• HA
• Ha)]}

Tra [1/: d EA 8(EA

• HA) 1/:

dE 8(E

• Ha)8(E
• HA
• Ha) J}

=

TrA { Tra [1/: d EA 8(EA

• HA) 1/:

dE 8(E- Ha)8(E

• EA- E) J}

d EA TrA[8(EA

• HA)] 1/:

dE Tr8[8(E

• Ha)]8(E
• EA
• E)

= 1/:

d E'.BwA(EA) 1/: dE WB(E)8(E- EA- E)

= fpop

d EA fpop dE WA(E'.B)w8(E)8(E- EA- E)

=fpE

dEAwA(E'.B)wa(E- E'.B).

Finite bath coupling

T HS HSB

γ

HB HS

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

Route I E . = ES = 〈HS〉 = TrS+B(HSe−βH) TrS+B(e−βH) Route II Z = TrS+B(e−βH) TrB(e−βHB) U = −∂lnZ ∂β ⇒ U = 〈H〉−〈HB〉B = ES +

• 〈HSB〉+ 〈HB〉−〈HB〉B
• For finite coupling E and U differ!

Strong coupling: Example

System: Two-level atom; “bath”: Harmonic oscillator H = ✏ 2z + Ω ✓ a†a + 1 2 ◆ + z ✓ a†a + 1 2 ◆ H∗ = ✏∗ 2 z + ✏∗ = ✏ + + 2 artanh ✓ e−βΩ sinh() 1 − e−βΩ cosh() ◆ = 1 2 ln ✓1 − 2e−βΩ cosh() + e−2βΩ (1 − e−βΩ)2 ◆ ZS = Tre−βH∗ FS = −kbT ln ZS SS = −@FS @T CS = T @SS @T

• M. Campisi, P. Talkner, P. H¨

anggi, J. Phys. A: Math. Theor. 42 392002 (2009)

Entropy and specific heat

Ω/✏ = 3 Ω/✏ = 1/3

Important UNSOLVED (open) Problems are: 1.) Quantum systems and discrete spectral parts: DoS becomes singular ===> a sum

• f delta-functions

!!! ??? !!! best smoothing procedure ???!!! 2.) Canonical ensemble: When is the Bolzmanfactor truly OK? 3.) Canonical ensemble and STRONG coupling: Quantum case: Canonical specific heat can now become negative (!) despite system being stable Classical case: Are *negative* canonical specific heat values possible?

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.

** 23 pages **

A QUESTION ?