Gert-Ludwig Ingold. Differences can be negative. Motivation. Specific heat and dissipation. Path I. Path II. Casimir effect.

Transcription

1 Gert-Ludwig Ingold I

2 I in collaboration with: Peter Hänggi Peter Talkner Universität Augsburg Acta Phys. Pol. B 37, 1537 (2006) New J. Phys. 10, (2008) Phys. Rev. E 79, (2009) Astrid Lambrecht Serge Reynaud Laboratoire Kastler Brossel, Paris Phys. Rev. E 80, (2009)

3 I I For the»mutual information«aficionados There will be differences of... specific heats C AB C B... and therefore of entropies S AB S B

4 II For the»system+environment«aficionados specific heat of a free particle I C/kB T

5 II For the»system+environment«aficionados specific heat of a free particle I C/kB T Does the free particle violate the third law of thermodynamics? P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006)

6 II (continued) We need an energy scale I

7 II (continued) I We need an energy scale 1 put particle into a box let the particle be free (only limited by the size of the observable universe) energy scale of K

8 II (continued) I We need an energy scale 1 put particle into a box 2 couple the particle to an environment new energy scale ħγ relevant quantity k BT ħγ

9 II (continued) I We need an energy scale 1 put particle into a box 2 couple the particle to an environment new energy scale ħγ relevant quantity k BT ħγ γ 0 corresponds to classical limit T Coupling to the environment makes the free particle more quantum! [somewhat in the spirit of J. R. Anglin, J. P. Paz, and W. H. Zurek, Deconstructing decoherence, PRA 97]

10 The problem H S I T

11 The problem H S γ H SB H B I T What actually do we mean by»specific heat of a dissipative system«?

12 Statistical physics 101 I β = 1 k B T U = lnz β density of states ρ Z = deρ(e)e βe canonical partition function Z S = F T = k BT lnz T internal energy U entropy S C = U T C = T S T specific heat C

13 From system energy to specific heat β = 1 k B T density of states ρ Z = deρ(e)e βe I U = lnz β system energy E = H S canonical partition function Z S = F T = k BT lnz T entropy S C = U T C = T S T specific heat C E

14 From partition function to specific heat I I β = 1 k B T density of states ρ canonical partition function Z = Tr S+B(e βh ) Tr B (e βh B ) U = lnz β internal energy U Z = deρ(e)e βe S = F T = k BT lnz T entropy S C = U T C = T S T specific heat C Z

15 An important difference E = H S = Tr S+B(H S e βh ) Tr S+B (e βh ) I Z = Tr S+B(e βh ) Tr B (e βh B ) U = lnz β I U = H H B B ] = E + [ H SB + H B H B B For finite coupling to the bath, E and U differ! There is no unique way to define a specific heat. P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006)

16 of a damped free particle I C E /kb ω D /γ = k B T /ħγ explicit results T : classical value k B /2 Damping constant γ determines temperature scale Third Law saved by coupling to the environment more damping makes the system more quantum

17 of a damped free particle I Partition function I C Z /kb 0.2 ω D /γ = explicit results k B T /ħγ already the leading high temperature corrections for C E and C Z differ

18 of a damped free particle I Partition function I C Z /kb 0.2 ω D /γ = explicit results k B T /ħγ already the leading high temperature corrections for C E and C Z differ The specific heat can become negative!? P. Hänggi, GLI, P. Talkner, New J. Phys. 10, (2008)

19 Differences make their appearance I reduced partition function Z = Z S+B Z B In order to obtain the entropy or the specific heat, one needs to take the logarithm. The reduced partition function leads to differences of entropies or of specific heats: C Z = C Z S+B CZ B i. e., how does the specific heat change if the system degree of freedom is coupled to the bath? The difference of two positive numbers can.

20 Origin of negative specific heat I M f B m I C Z /kb f B m k B T/ hω GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009)

21 Origin of negative specific heat I M f B m I C Z /kb f B m k B T/ hω GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009) Coupling of a degree of freedom to an environment can lead to a reduction of the specific heat.

22 I

23 I finite temperature finite permittivity geometry surface roughness finite thickness

24 I finite temperature finite permittivity geometry surface roughness finite thickness

25 From ideal to real mirrors I ideal mirrors ε(ω) =

26 From ideal to real mirrors I ideal mirrors plasma model ε(ω) = ε(ω) = 1 ω2 P ω 2

27 From ideal to real mirrors I ideal mirrors plasma model Drude model ε(ω) = ε(ω) = 1 ω2 P ω 2 ε(ω) = 1 ω 2 P ω(ω + iγ)

28 From ideal to real mirrors I ideal mirrors plasma model Drude model ε(ω) = ε(ω) = 1 ω2 P ω 2 negative entropy! ε(ω) = 1 ω 2 P ω(ω + iγ)

29 Length scales I ε(ω) = 1 λ P µm ω 2 P ω(ω + iγ) λ P = 2πc ω P λ T = hc k B T λ T 7.6 µm λ γ 100 µm L ( ) ħγ S TE f k B T

30 Casimir entropy I (L 2 /A)S/kB Lγ/c = k B T L/ hc GLI, A. Lambrecht, S. Reynaud, Phys. Rev. E 80, (2009)

31 I Dissipation can save the Third Law of thermodynamics. For nonnegligible coupling to the bath, specific heat is not uniquely defined. The reduced partition function can lead to negative values of the (difference of) specific heat(s). This scenario finds an application in the. References: P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006) P. Hänggi, GLI, P. Talkner, New J. Phys. 10, (2008) GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009) GLI, A. Lambrecht, S. Reynaud, Phys. Rev. E 80, (2009)

32 I Dissipation can save the Third Law of thermodynamics. For nonnegligible coupling to the bath, specific heat is not uniquely defined. The reduced partition function can lead to negative values of the (difference of) specific heat(s). This scenario finds an application in the. References: P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006) P. Hänggi, GLI, P. Talkner, New J. Phys. 10, (2008) GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009) GLI, A. Lambrecht, S. Reynaud, Phys. Rev. E 80, (2009)

33 I Dissipation can save the Third Law of thermodynamics. For nonnegligible coupling to the bath, specific heat is not uniquely defined. The reduced partition function can lead to negative values of the (difference of) specific heat(s). This scenario finds an application in the. References: P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006) P. Hänggi, GLI, P. Talkner, New J. Phys. 10, (2008) GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009) GLI, A. Lambrecht, S. Reynaud, Phys. Rev. E 80, (2009)

34 I Dissipation can save the Third Law of thermodynamics. For nonnegligible coupling to the bath, specific heat is not uniquely defined. The reduced partition function can lead to negative values of the (difference of) specific heat(s). This scenario finds an application in the. References: P. Hänggi, GLI, Acta Phys. Pol. B 37, 1537 (2006) P. Hänggi, GLI, P. Talkner, New J. Phys. 10, (2008) GLI, P. Hänggi, P. Talkner Phys. Rev. E 79, (2009) GLI, A. Lambrecht, S. Reynaud, Phys. Rev. E 80, (2009)

35 From system energy to specific heat with C E k B = x 1x 2 x 1 x 2 [ x2 ψ (x 2 ) x 1 ψ (x 1 ) ] 1 2 x 1,2 = ħβω D 4π ( 1 ± ) 1 4γ ω D I High temperature expansion C E k B = 1 2 ħ2 γω D 24(k B T) 2 + O(T 3 ) Low temperature expansion C E = π ( ) k B T k B 3 ħγ 4π3 kb T 3 ( 1 2 γ ) + O(T 5 ) 15 ħγ ω D back

36 Partition function of the damped free particle I Z = L ħ β ( ) 2πm 1/2 n=1 ν n ν n + ˆγ(ν n ) back

37 From partition function to specific heat with C Z ( ) 2 ( ) = x1 2 k ψ (x 1 ) + x2 2 ħβωd ψ (x 2 ) ψ ħβωd 1 B 2π 2π 2 x 1,2 = ħβω D 4π ( 1 ± ) 1 4γ ω D I High temperature expansion C Z k B = 1 2 ħ2 γω D 12(k B T) 2 + O(T 3 ) Low temperature expansion C Z = π ( k B T 1 γ ) ( 4π3 kb T k B 3 ħγ ω D 15 ħγ ) 3 [ 1 3 γ ( ) γ 3 ] + O(T 5 ) ω D ω D back