Thermodynamics, Statistical Mechanics, and Quantum mechanics with some flavor of information theory

Transcription

1 Beyond IID, July 28, 2017 The second law of thermodynamics from the view points of Thermodynamics, Statistical Mechanics, and Quantum mechanics with some flavor of information theory Hal Tasaki partly with Sheldon Goldstein and Takashi Hara

2 Thermodynamics

3 what is thermodynamics? quantitatively exact macroscopic phenomenological theory about possible transitions between equilibrium states energy transfer associated with transitions The second law of thermodynamics is, without a doubt, one of the most perfect laws in physics. Any reproducible violation of it, however small, would bring the discoverer great riches as well as a trip to Stockholm. (Lieb and Yngvason) formulated entirely within macroscopic description without references to microscopic world a crucial guide in the revolution from classical to quantum mechanics

4 what is thermodynamics? Lieb and Yngvason, The physics and mathematics of the second law of thermodynamics (1997) rigorous operational formulation with a deep physical insight S(X) =sup{λ : (X Γ, λz 1 ) (X, λz 0 )} modern textbooks from fully operational points of view Hal Tasaki and Glenn Paquette Theromdynamics: a modern point of view (to be published in 2018?) 5

5 description of equilibrium states any isolated macroscopic system approaches an equilibrium state after a long time no changes, no flows an equilibrium state is fully characterized by (U, X) U X energy set of extensive variables, such as X =(V,N) i =1, 2,...,n M i macroscopic quantities equilibrium values M i (U, X) volume amount of substance

6 adiabatic operation enclose the system in a thermally insulating wall change X to X 0 mechanically wait for a while (U, X) mechanical work W (U 0,X 0 ) adiabatic operation (U, X)! (U 0,X 0 ) the total work done to the agent W = U 0 U the first law of thermodynamics (energy conservation law)

7 second law entropy S(U, X) concave in U, X, increasing in U second law (entropy principle) adiabatic operation(u, X)! (U 0,X 0 )is possible if and only if S(U, X) apple S(U 0,X 0 ) quasi-static (reversible) operation if we change X indefinitely slowly (U, X)! (U 00,X 00 ) we have S(U, X) =S(U 00,X 00 ) thus (U 00,X 00 )! (U, X) is also possible

8 Passivity or Planck s principle cyclic adiabatic operation the parameters return to the initial values (U, X) (U 0,X) if (U, X)! (U 0,X) is possible then U apple U 0 the most essential form of the send law other formulations of the second law can be derived from Planck s principle Kelvin's principle: No perpetual mobile of the 2nd kind illustration by Chisato Naruse 10

9 Statistical mechanics

10 what is statistical mechanics? based on microscopic mechanical description of macroscopic systems probabilistic models which are consistent with thermodynamics, and reproduce macroscopic properties of equilibrium states may not be exactly equal to states which are realized there are many distributions (i.e., probabilistic models) which are equivalent (from a macroscopic point of view) microcanonical, canonical, grand canonical,

11 microscopic model model of a macroscopic system with volume V example: N particles in volume V Hilbert space H sys Hamiltonian Ĥ sys energy eigenstates Ĥ sys j i = E j j i h j j i =1 j =1, 2,... the number of states (the # of j s.t. E j apple U) behaves as (U) =exp[v (U/V, N/V )+o(v )] can be proved in many systems (u, ) entropy density concave in (u, ), increasing in u

12 standard distributions microcanonical distribution ˆP [U <Ĥsys apple U + V ] mc = (U + V ) (U) U + V U canonical distribution very small constant (indep. of V) can = exp[ Ĥ sys ] Z( ) F (T ; X) = 1 log Z( ) inverse temperature =(k B T ) 1 is chosen so that Tr[Ĥsys ˆ can ]=U

13 equivalence of ensembles microcanonical distribution ˆP [U <Ĥsys apple U + V ] mc = (U + V ) (U) canonical distribution can = exp[ Ĥ sys ] Z( ) F (T ; X) = 1 log Z( ) equivalence of thermodynamic functions V 1 S(U, X) ' V 1 k B log{ (U + V ) (U)} ' V 1 inf {U F (T ; X)}/T T equivalence of expectation values M i (U, X) ' Tr[ ˆM i mc ] ' Tr[ ˆM i can ] proved in general proved in limited cases 15

14 Statistical mechanical derivation of the second law traditional approach

15 traditional modeling of operation the operation by the agent is modeled by a Hamiltonian Ĥ sys (t)(with t 2 [0, ] ) whose parameters vary according to a fixed protocol in such a way that Ĥ sys (0) = Ĥsys( ) =Ĥsys the system evolves according to the corresponding R unitary time evolution Û = t-exp[ i 0 dtĥsys(t)] initial state init = mc or can with energy U final state fin = Û initû U 0 =Tr[Ĥsys fin ] second law with energy U apple U 0

16 derivation with relative entropy initial state (canonical distr.) and the final state init = can = exp[ Ĥ sys ] Z( ) fin = Û initû relative entropy D(ˆ fin kˆ init )=Tr[ˆ fin log ˆ fin ] Tr[ˆ fin log ˆ init ] =Tr[ˆ init log ˆ init ] Tr[ˆ fin log ˆ init ] = Tr[Ĥsys ˆ init ]+Tr[Ĥsys ˆ fin ] U U 0 D(ˆ fin kˆ init ) 0 U apple U 0 2nd law

17 derivation with majorization Ĥ sys j i = E j j i initial and final probabilities E j apple E j+1 p j = h j ˆ init j i p 0 j = h j ˆ fin j i = h j Û ˆ initû j i = P k T jkp k transition matrix (doubly T jk = h j Û ki 2 stochastic) Pj T jk = P k T jk =1 final energy U 0 = P is bounded j E jp 0 j = P as j,k E jt jk p k Pj E jp # j where p # j = p (j) p # j p # j+1

18 derivation with majorization final energy is bounded as U 0 = P j E jp 0 j = P j,k E jt jk p k Pj E jp # j where p # j = p (j) p # j p # j+1 canonical initial state init = can p j = e we have E j /Z( ) p # j = p j U 0 Pj and hence E jp j = U microcanonical initial state init = mc p # j 6= p j but we can make use of the macroscopic nature of the system to show U 0 U

19 0 = V (U + V ) V (U) derivation with majorization p j = h j ˆ mc j i V (E) =exp V (E/V )+o(v ) p j p # j 0 not enough dimensions to lower the energy! U V (U)

20 second law in terms of probability initial state init = mc or can with energy U final state fin = Û initû theorem: there exists a constant, and we have Prob fin [Ĥsys apple U] :=Tr ˆP [ Ĥ sys apple U] ˆ fin apple e 0V for sufficiently large V 0 when V is large enough, it essentially does not happen that the final energy is less than U (the second law of thermodynamics) Lenard 1978, (Tasaki 2000)

21 the unitarity problem mechanical work W thermodynamics: the agent knows the amount of work quantum mechanics: in order to know the amount of work the agent must continuously make measurement of the system the time evolution of the system cannot be unitary! Hayashi, Tajima 2016

22 summary of this part we have reviewed the traditional derivation of the second law based on the unitary time evolution this treatment suffers from the unitarity problem 22

23 Statistical mechanical derivation of the second law hybrid setting

24 dichotomy (unitarity problem) one can have either necessary for the standard thermodynamics! the agent CAN determine the amount of work the time evolution of the system is NOT unitary or the time evolution of the system IS unitary the agent CANNOT determine the amount of work suitable for thermodynamics for small quantum systems?? the dichotomy has been formulated as a theorem by Hayashi and Tajima

25 dichotomy (unitarity problem) the agent CAN determine the amount of work the time evolution of the system is NOT unitary we shall treat macroscopic systems in this setting we cannot use the traditional setting with timedependent Hamiltonian and unitary timeevolution hybrid setting

26 Skrzypczyk, Short, Popescu 2014, Frenzel, Jennings, Rudolph 2014, Malabarba, Short, Kammerlander 2015 hybrid setting model everything quantum mechanically (thermodynamic) system clockwork agent apparatus mechanical system which operates on the system supplies energy macroscopic system the total energy of the system + apparatus is conserved system + apparatus is isolated from the external world, and evolves autonomously with a timeindependent Hamiltonian (c.f. clock + weight )

27 apparatus mechanical system that operates on the system Hilbert space H ap Hamiltonian Ĥ ap anything D ap := dim[h ap ] < 1 not too large, independent of V a rough (over)estimate typical mass M 1 kg length L 10 m artificial (but harmless) cutoff velocity v max 10 4 m/s degrees of freedom n 100 Mvmax L n. e 10 4 e 1023 D ap h

28 general hybrid setting (thermodynamic) system apparatus mechanical system which operates on the system supplies energy Hilbert space H ap Hilbert space H sys Hamiltonian Ĥ sys Hamiltonian Ĥ ap D ap := dim[h ap ] the whole system H tot = H sys H ap evolves autonomously with time-independent Hamiltonian Ĥ tot = Ĥsys + Ĥap + Ĥint arbitrary interaction

29 2nd law in hybrid setting you can never raise the weight and cool the gas!! U U 0 U<U 0 there is nothing like a SUPER-elastic collision!! v>v 0 v 0 v apparatus system energy flow

30 collision of a macroscopic ball the ball which consists of a macroscopic number of small particles collides with a wall system Ĥ sys apparatus Ĥ ap internal degrees of freedom of the ball the center of mass motion of the ball Ĥ int interaction between the wall and the particles microscopic model of inelastic collision (classical case Tasaki 2006, Maes-Tasaki 2007)

31 theorem (2nd law) initial state ˆ init =ˆ mc ' 0 ih' 0 arbitrary microcanonical in [U, U + V ] final state ˆ fin = e iĥtot ˆ init e iĥtot projection onto the subspace with Ĥ sys apple U ˆP appleu = ˆP [Ĥsys apple U] ˆ1 ap theorem: there exists a constant 1 > 0 and one has Prob fin [Ĥsys apple U] :=Tr[ˆP appleu ˆ fin ] apple e 1V the final energy of the system (almost certainly) does not exceed U when V is large

32 the essence of the proof There is not enough dimension to lower the energy V (E) =exp V (E/V )+o(v ) ˆ mc 0 dim U V (U)D ap dim 0 ' 0 i ' 1 i ' 0 i ' 1 i D ap =2

33 summary of this part we have proved the second law in the hybrid setting when the initial state of the (thermodynamic) system is described by an equilibrium ensemble we rely essentially on the macroscopic nature of the system D ap V (U + V ) e const.v V (U) similar to clock+weight approach, but is very rough (since only macroscopic systems are treated) Skrzypczyk, Short, Popescu 2014 Frenzel, Jennings, Rudolph 2014 Malabarba, Short, Kammerlander

34 Quantum mechanics

35 motivation what are actual states realized in macroscopic quantum systems in equilibrium? can never be one of the standard equilibrium distributions (e.g., canonical, microcanonical) many states (including pure states) represent equilibrium (typicality) can we derive the second law starting from a pure initial state?

36 Typicality of equilibrium states All the micro-states with energy U A great majority of microscopic states with energy U look identical from the macroscopic point of view thermal equilibrium = common properties shared by these majority of states A single pure state may represent equilibrium! von Neumann 1929, Llyoid 1988, Sugita 2006, Popescu, Short, Winter 2006 Goldstein, Lebowitz, Tunulkam Zanghi 2006, Reimann 2007

37 Quantum mechanical derivation of the second law hybrid setting

38 initial state initial state of the system a normalized pure init i = P state j c j j i with energy almost U P j c j 2 =1 small constant c j 6=0only when E j 2 [U, U + V ] moderate energy distribution effective dimension D e := P j c j 4 1 Linden, Popescu, Short, Winter 2009 V (U + V /2) U + V most states satisfy this because U max D e = V (U + V ) V (U) V (U + V /2)

39 initial state initial state of the apparatus arbitrary pure state ' 0 i2h ap initial state of the whole system initi = init i ' 0 i ' 0 i init i

40 2nd law? there can be a miracle in which the energy of the system in the final state fini = e iĥtot init i ' 0 i becomes much lower than U U>U 0 U U 0 But this requires an extraordinarily fine tuning, and must be extremely unstable

41 random initial phase initial state of the system init i = P j c j j i introduce =( j ) j=1,2,... j 2 [0, 2 ) and let init i = P j ei j c j j i lack of information one does not know completely about the phases have complete control of the phases initial state initi = initi ' 0 i lack of controllability final state fini = e iĥtot initi ' 0 i

42 theorem (2nd law) Ĥ sys apple U projection onto the subspace with ˆP appleu = ˆP [Ĥsys apple U] ˆ1 ap theorem: there exists a constant 2 > 0 Suppose that one chooses each phase j 2 [0, 2 ) randomly. Then with probability one has Prob fin[ĥsys apple U] :=h fin ˆP appleu 1 e 2V fini applee 2V for most choice of, the final energy of the system (almost certainly) does not exceed U when V is large

43 summary of this part we have proved the second law in the hybrid setting when the initial state of the (thermodynamic) system is a pure state with moderate energy distribution the theorem requires the introduction of random phases, which represents the lack of knowledge/controllability of the phases the theorem agin relies essentially on macroscopic nature of the system

44 Summary we have discussed (or derived) the second law (Planck s principle) in the contexts of thermodynamics statistical mechanics + quantum dynamics quantum mechanics our approach essentially depends on macroscopic nature of the system can we develop universal and useful thermodynamics for small quantum systems? illustration by Chisato Naruse Skrzypczyk, Short, Popescu 2014 Frenzel, Jennings, Rudolph 2014 Malabarba, Short, Kammerlander 2015 Horodecki, Oppenheim 2013 Brandao, Horodecki, Oppenheim, Wehner 2015 Tajima, Hayashi 2015 Ito, Hayashi 2016

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