Beyond the Second Law of Thermodynamics C. Van den Broeck R. Kawai J. M. R. Parrondo Colloquium at University of Alabama, September 9, 2007
The Second Law of Thermodynamics There exists no thermodynamic transformation whose sole effect is to extract a quantity of heat from a given heat reservoir and to convert it entirely into work. William Thomson Lord Kelvin (822 873) They state what cannot happen but do not tell you what will actually happen! There exists no thermodynamic transformation whose sole effect is to extract a quantity of heat from a colder reservoir and to deliver it to a hotter reservoir. Rudolf Clausius (822 888)
Entropy and the Second Law Heat Bath S 0 S Q T Q T Isolated Systems Closed Systems No exchange of energy or matter between the system and the environment is allowed. Energy exchange is allowed but not matter exchange. Second Law Time's Arrow!
Sir Arthur Eddington (882-944) The law that entropy always increases holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equation then so much the worse for Maxwell's equations... 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. (928) The second law of thermodynamics is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of the basic concepts, it will never be overthrown. (949) Albert Einstein (879-955)
Why is the second law an inequality? S Q = T Holy Grail of Statistical Mechanics? 0 S=S r S i Q T reversible entropy change Sr = irreversible entropy production S i 0 Ludwig E. Boltzmann (844-906)
Second Law with Work U=W Q (First Law of Thermodynamics) F = U T S (Helmholtz Free Energy) W F =T S Q= Holy Grail 0 W =W rev W dis reversible work W rev = F dissipative work W dis 0
Holy Grail Revealed in Phase Space F q, p,t W F =k B T F q, p,t ln dq dp B q, p,t =k B T D F B
A Non-Equilibrium Process: Time-Dependent Hamiltonian t0 equilibrium T H q, p ; 0 W t t H q, p ; t H q, p ; Forward Process Q Q' Backward Process H q, p ; 0 T H q, p ; H q, p ; t W' equilibrium
Example T 0 F W L Forward T NonEquilibrium T L2 Backward V L L2 F W 0
Phase Space Trajectory and Density 6 N -dimension phase space q, p [q t, p t ] t p q= q, q 2,, q 3N position p= p, p 2,, p 3N momentum [q t, p t ]=phase trajectory t0 q, p,t =probability density q 0, p0 q Liouville Theorem q 0, p 0, t = q t, p t,t = q, p, t Microscopic Time Reversibility q 0, p 0 q, p q, p q 0, p 0 Joseph Liouville (809-882)
Gibbs Entropy S= k B q, p ln q, p dq dp J. Willard Gibbs (839-903) S t = k B q, p, t ln q, p, t dq dp Invariant under Hamiltonian Dynamics S t 0 =S t =S t q, p,t ln q, p, t dq dp is invariant!
Thermal Equilibrium Equilibrium Density eq q, p = exp [ H q, p ] Z Z= exp[ H q, p ]dq dp (partition function) eq q, p = eq q p (detailed balance) H q, p = k B T ln Z k B T ln eq q, p
Definition of Work q, p W q 0, p 0 t0 t W q 0, p 0 =H q, p ; H q 0, p 0 ; 0 Statistical Average W = q 0, p 0 ; t 0 W q o, p 0 dq 0 dp0 = q 0, p 0 ; t 0 [H q, p ; H q 0, p 0 ; 0 ]
Proof W = q 0, p 0 ; t 0 [H q, p ; H q0, p0 ; 0 ] = kt F q, p, t ln B q, p, t dq dp kt F q0, p0, t 0 ln F q 0, p0, t 0 dq 0, dp 0 kt ln Z 0 /Z = kt F q, p, t ln B q, p,t dq dp kt F q, p,t ln F q, p,t dq dp F [ ] F q, p, t W F =kt F q, p,t ln dq dp=kt D F B B q, p,t
Relative Entropy (Kullback-Leibler distance) D = x ln x dx x x 0, x 0 ; x dx = x dx = D is a `distance` between two densities. D 0, D =0 iff x = x exp[ D ] is a measure of the difficulity to statistically distinguish two densities. (Stein's lemma) D D if and have less information than and
Relative Entropy: Exercise with Dice normal p = biased I q = biased II 6 p 2= 6 p 3= 6 p 4= 6 q 2= q 3= q 4= 3 2 2 2 95 r = r 2= r 3= r 4= 00 00 00 00 6 pi D p q = pi ln =0.63 qi i = 6 pi D p r = p i ln =2.054 ri i = p 5= 6 p 6= 6 q 6= 6 4 q 5= r 6= 00 00 q 5=
Find which dice you have by rolling it N times p vs. q (good one vs. slightly biased one) N D p q P err N =e p vs. r, P err 0 =0.96, P err 20 =0.04, perr 50 =0.00028 (good one vs. extremely biased one) N D p r P err N =e 4 9, P err 5 =0.3 0, P err 0 =0. 0 You will notice already here.
Relative Entropy and Reduced Information normal dice p odd =p p 3 p 5=, 2 7 odd =q q 3 q 5 =, biased dice q 2 p even =p 2 p 4 p 6= 5 q even =q 2 q 4 q 6 = 2 D p q =0.63 p odd p even D p q = p odd ln p even ln =0.04 q odd q even D p q D p q 2
Dissipation and Time's Arrow [ ] F q, p,t W F =kt F q, p, t ln dq dp=kt D F B B q, p, t D F B 0 If F = B, Second Law D F B =0 No Dissipation Dissipation is a quantitative measure of Irreversibility (time's arrow)!.
Slow Expansion Forward q 0, p 0 Backward q 0, p 0 t =0 t =0 t= /2 t= / 2 t= t = q, p q, p No Dissipation
Rapid Expansion Forward q 0, p 0 Backward q 0, p0 t =0 t =0 t= /2 t= /2 t= t= q, p q, p
Which direction is the triangle moving?
Jarzinski equality and Crooks Theorem [ ] F q, p,t W dis =kt F q, p,t ln q, p,t dq dp B Work at a phase point F q, p,t W dis q, p,t =kt ln B q, p, t Crooks theorem (can be negative) B q, p,t exp[ W dis q, p, t ]= F q, p,t Jarzynski equality exp[ W dis ] = F q, p,t exp[ W dis q, p,t ]dq dp=
Coarse Graining Devide the whole phase space into N subsets j j = N j j F t = F q, p, t dq dp ; B t = B q, p,t dq dp xj x j W j F kt ln j F j B j j t0 t t
j j W F kt D F B N where D Fj Bj = Fj ln j = j F Bj Since we don't have full information of the phase densities, we can have only a lower bound. If we have no information at all (N=), then D F B =0 W F 2nd law!
Overdumped Brownian Particle in a Harmonic Potential x, t = x, p, x, p, x 2, p 2,,t dp dx dp Forward Process Backward Process Heat Bath W F D F x, t B x,t
Application: Physics and Information Szilard found a relation between physics and information. bit =k B l o g 2 Leó Szilárd (898-964) Landauer principle The erasure of one bit of information is necessarily accompanied by a dissipation of at least kbt log 2 heat. Information can be obtained without dissipation of heat. Q k B T log 2 Ralf Landauer (929-999) i i2 OR o
Szilard's Engine T Q W =k B T ln 2 Contradiction to 2nd Law? T T W Q kt ln2 T T
Brownian Engine (Backword Process) W = kt ln 2 b a d c
Brownian Computer (Forward Process) a b W = kt ln 2 Recording Erasure d c Restore-to-One Procedure: d a b c d
Coarse Grained Measurement L R PF x PB x 2 L - R L [ ] R P F R W R ln =k B T ln 2 k B T ln P B R
For quantum systems von Neumann Entropy : S= k Tr ln ] W dis =kt [ Tr F ln F Tr F ln B
Conclusion F q, p,t W F =k B T F q, p, t ln dq dp B q, p, t =k B T D F B An exact expression of dissipation is obtained. Now the second law of thermodynamics is an equality! Dissipation is a direct measure of irreversibility (time's arrow). Even when full information is not available, the formula provides a lower bound of the dissipation The relation between information and physical processes is unambiguously formulated. The Landauer principle is proven.