TOWARD THERMODYNAMICS FOR NONEQUILIBRIUM STEADY STATES OR TWO TWISTS IN THERMODYNAMICS FOR NONEQUILBRIUM STEADY STATES HAL TASAKI WITH T.S.KOMATSU, N.NAKAGAWA, S.SASA PRL 100, 230602 (2008) and papers in preparation KEK Oct. 30, 2009
ULTIMATE GOAL (WHICH MAY NOT BE REACHABLE) CONSTRUCT A UNIVERSAL STATISTICAL MECHANICS FOR NONEQULIBRIUM STEADY STATE (NESS) TENTATIVE GOAL (WHICH IS STILL NOT AT ALL EASY) CONSTRUCT A SENSIBLE OPERATIONAL THERMODYNAMICS FOR NESS (AS A GUIDELINE FOR STATISTICAL MEHCANICS)
THERMODYNAMICS AND STATISTICAL MECHANICS THERMODYNAMICS A SET OF UNIVERSAL (AND EXACT!) RELATIONS FOR VARIOUS PROPERTIES, HEAT/ENERGY EXCHANGE IN MACROSCOPIC SYSTEMS IN EQUILIBRIUM STATISTICAL MECHANICS UNIVERSAL FRAMEWORK WHICH ENABLES ONE TO COMPUTE ANY MACROSCOPIC PROPERTIES OF A GIVEN SYTEM STARTING FROM ITS MICROSCOPIC (MECHANICAL ) DESCRIPTION HISTORICAL NOTE: THERMODYNAMICS WAS AN INDISPENSABLE GUIDE WHEN THE EQUILIBRIUM STATISTICAL MECHANICS WAS CONSTRUCTED
BASIC SETUP
TYPICAL SYSTEM CLASSICAL MECHANICAL SYSTEM WITH FINITE BOX Λ r i Λ R 3 p i R 3 POSITION MOMENTUM Γ = (r 1,..., r N ; p 1,..., p N ) Λ N PARTICLES IN A OF THE i-th PARTICLE pi = mv i T 1 T 2 THE SYSTEM IS ATTACHED TO TWO HEAT BATHS
TIME EVOLUTION USUAL NEWTON EQUATION m d2 r i (t) dt 2 = grad i V (r 1 (t),..., r N (t)) V (r 1,..., r N )= i V (ν) ext (r i )+ i<j v(r i r j ) ν (CONTROLLABLE) PARAMETER (E.G., THE VOLUME) STOCHASTIC TIME EVOLUTION AT THE WALLS THERMAL WALL, LANGEVIN DYNAMICS (ONLY NEAR THE WALLS), ETC. Λ T 1 T 2 WE ONLY NEED LOCAL DETAILED BALANCE CONDITION
THERMAL WALL v in A PARTICLE WITH ANY INCIDENT VELOCITY IS BOUNCED BACK WITH A RANDOM VELOCITY v out WITH THE PROBABILITY DENSITY p T (v out )=Av out x exp [ m vout 2 2kT v out =(v out x T v in,vy out,vz out ) k v out x T v out ] x > 0 vy out A = 1 2π,v out z BOLTZMANN CONSTANT TEMPERATURE OF THE WALL ENERGY (HEAT) TRANSFERED FROM THE BATH TO THE SYSTEM ( m kt ) 2 R q = m 2 vout 2 m 2 vin 2
EQUILIBRIUM STATE AND CLAUSIUS RELATION
EQUILIBRIUM SYSTEM SET T 1 = T 2 = T T T AS t THE SYSTEM APPROACHES THE UNIQUE EQUILIBRIUM STATE WITH THE PROBABILITY DENSITY ρ eq (T ) (Γ) = 1 Z(T ) exp[ H(Γ) ] kt k BOLTZMANN CONSTANT HAMILTONIAN H(Γ) = N i=1 p i 2 2m + V (r 1,..., r N )
EQUILIBRIUM THERMODYNAMIC OPERATION START FROM EQUILIBRIUM AND SUDDENLY CHANGE THE TEMPERATURE FROM T TO THE PARAMETER FROM TO EQUILIBRIUM T T T T T T SUDDEN CHANGE ν T = T + T ν = ν + ν ν ν ν RELAXATION TO NEW EQUILIBRIUM Q ENERGY (HEAT) TRANSFERED TO THE SYTEM FROM THE BATHS DURING THE RELAXATION PROCESS
CLAUSIUS RELATION Q ENERGY (HEAT) TRANSFERED TO THE SYTEM FROM THE BATHS DURING THE RELAXATION PROCESS T T T T T T CLAUSIUS RELATION (UNIVERSAL RELATION) S(T, ν ) S(T,ν) = Q = O( T )+O( ν) T + O( 2 ) AMOUNT OF THE CHANGE S(T,ν) WHERE THE THERMODYNAMIC ENTROPY HAS A MICROSCOPIC REPRESENTATION IN TERMS OF THE GIBBS- SHANNON ENTROPY S(T,ν) =S Sh [ρ eq ( )] = k dγ ρ eq (Γ) log ρ eq (Γ)
NONEQUILIBRIUM STEADY STATES AND OPERATION
SET T 1 T 2 NONEQUILIBRIUM STEADY STATE (NESS) T 1 T 2 T 1 T 2 AS t THE SYSTEM IS EXPECTED TO APPROACH A UNIQUE STATIONARY STATE = NONEQUILIBRIUM STEADY STATE (NESS) (PROVIDED THAT THE DEGREE OF NONEQULIBRIUM IS SMALL) EQUILIBRIUM STATE: NESS: FIX THE PARAMETERS ν NO MACROSCOPIC CHANGES NO MACROSCOPIC FLOWS NO MACROSCOPIC CHANGES NONVANISHING MACROSCOPIC FLOW OF ENERGY OR MATTER
NONEQUILIBRIUM STEADY STATE (NESS) T 1 NO! FOR THE MOMENT T 2 ρ (T 1,T 2,ν) ss (Γ) PROBABILITY DENSITY IN THE NESS WITH T 1,T 2, ν IS THERE STATISTICAL MECHANICS FOR NESS, I.E. A UNIVERSAL MATHEMATICAL FRAMEWORK FOR DETERMINING ρ ss FOR A GNERAL SYSTEM? BEFORE STAT. MECH. WE WILL LOOK FOR THERMODYNAMICS FOR NESS (WHICH IS ALSO UNKNOWN FOR THE MOMENT)
OPERATION TO NESS START FROM NESS AND SUDDENLY CHANGE THE TEMPERATURES FROM T TO T j j = T j + T j (j =1, 2) THE PARAMETER FROM TO ν ν = ν + ν NESS NESS T 1 T T 1 T 2 2 T 1 T 2 ν ν ν SUDDEN CHANGE RELAXATION TO NEW NESS CAN THERE BE A RELATION LIKE THE CLAUSIUS RELATION? NO! (NAIVELY SPEAKING) THERE IS A CONSTANT FLOW OF ENERGY (HEAT) THERE ARE TWO TEMPERATURES
EXTENDED CLAUSIUS RELATION AND THE ENTROPY FOR NESS THE FIRST ORDER RELATIONS
J ss AT TIME NESS EXCESS HEAT j (T 1,T 2, ν) HEAT (ENERGY) CURRENT FROM BATH j TO THE SYSTEM IN THE STEADY STATE WITH T 1,T 2, ν J j (t) AVERAGED HEAT (ENERGY) CURRENT FROM BATH j t NESS TOTAL HEAT TRANSFER IS DIVERGENT IN TIME T 1 T T 1 T 2 2 T 1 T 2 INTRINSIC HEAT TRANSFER CAUSED BY THE CHANGE (EXCESS HEAT) IS FINITE Jj ss (T 1,T 2, ν) Jj ss (T 1,T 2, ν ) SUDDEN CHANGE J j (t) RELAXATION TO NEW NESS t
EXCESS HEAT WE THUS DEFINE THE EXCESS HEAT (IN THE CASE OF STEP OPERATION) AS Q ex j = 0 dt{ J j (t) J ss j (T 1,T 2, ν )} Jj ss (T 1,T 2, ν) Jj ss (T 1,T 2, ν ) J j (t) Q ex j 0 t THE IDEA OF EXCESS HEAT IS (PROBABLY) DUE TO OONO-PANICONI 98 (SEE ALSO HATANO-SASA 01)
EXTENDED CLAUSIUS RELATION THERE EXISTS ENTROPY S(T 1,T 2 ; ν) S(T, T ; ν) =S(T ; ν) FOR NESS SUCH THAT, AND ONE HAS S(T 1,T 2; ν ) S(T 1,T 2 ; ν) = Qex 1 + Qex 2 T 1 T 2 +O(ɛ 2 )+O( 2 ) THIS HOLDS FOR NESS IN A LARGE CLASS OF SYSTEMS DEGREE OF NONEQUILIBRIUM ɛ = max{o(t 1 T 2 ),O(T 1 T 2)} AMOUNT OF CHANGE ESSENTIALLY THE SAME RESULT WAS SHOWN BY D. RUELLE (2003) IN (SIMPLER) SYSTEMS WITH ISOKINETIC THERMOSTAT
1st TWIST MICROSCOPIC EXPRESSION FOR THE ENTROPY THE NONEQUILIBRIUM ENTROPY IS RELATED TO THE PROBABILITY DENSITY BY S(T 1,T 2 ; ν) =S sym [ρ (T 1,T 2 ;ν) ss ( )] WITH THE SYMMETRIZED SHANNON ENTROPY S sym [ρ( )] := k dγ ρ(γ) log ρ(γ) ρ(γ ) TIME REVERSAL STATE Γ = (r 1,..., r N ; p 1,..., p N ) Γ =(r 1,..., r N ; p 1,..., p N )
ABOUT THE SYMMETRIZED SHANNON ENTROPY S sym [ρ( )] := k TIME REVERSAL dγ ρ(γ) log ρ(γ) ρ(γ ) STATE Γ = (r 1,..., r N ; p 1,..., p N ) Γ =(r 1,..., r N ; p 1,..., p N ) IF ρ(γ) = ρ(γ ) ONE HAS S sym [ρ( )] = S Sh [ρ( )] := k dγ ρ(γ) log ρ(γ) TWO ARE THE SAME IF THERE ARE NO MOMENTA
DERIVATION THE PUBLISHED DERIVATION IS SOMEWHAT INVOLVED, BUT WE NOW HAVE AN EFFICIENT DERIVATION LINEAR RESPONSE REPRESENTATION OF THE STEADY STATE PROBABILITY DISTRIBUTION ρ ss (Γ) = exp [ C + T dt 0 j=1,2 + J j (t) T j Γ eq + O(ɛ 2 ) THE DEFINITION OF SYMMETRIZED SHANNON ENTROPY ] EXTENDED CLAUIUS RELATION
INTEGRATING THE EXTENDED CLAUSIUS RELATION
INTEGRATED RELATION CHANGE THE TEMPERATURES AND THE PARAMETER CONTINUOUSLY AND SLOWLY IN TIME LONG TIME T 1 (t),t 2 (t), ν(t) t [0, T ] T j (0) = T j,t j (T )=T j, ν(0) = ν, ν(t )=ν (INTEGRATED) EXTENDED CLAUSIUS RELATION S(T 1,T 2; ν ) S(T 1,T 2 ; ν) { Q ex = 1 (t) + Qex 2 (t) T steps 1 (t) T 2 (t) + O(ɛ 2 )+O( 2 ) }
INTEGRATED RELATION (INTEGRATED) EXTENDED CLAUSIUS RELATION S(T 1,T 2; ν ) S(T 1,T 2 ; ν) { Q ex = 1 (t) + Qex 2 (t) T steps 1 (t) T 2 (t) = T 0 θ(t) dt + O(ɛ 2 ) + O(ɛ 2 )+O( 2 ) } EXCESS ENTROPY PRODUCTION RATE (IN THE BATHS) θ(t) = j=1,2 J j (t) J ss j (T 1(t),T 2 (t); ν(t)) T j (t)
OPERATIONAL DETERMINATION OF ENTROPY FIX T 1 (t) =T 1 AND ν(t) =ν AND CHANGE WE THEN HAVE T 2 (t) S(T 1,T 2 ; ν) S(T 1,T 1 ; ν) FROM T 2 (0) = T 1 TO T 2 (T )=T 2 = O(T 2 T 1 )=O(ɛ) = T EQUILIBRIUM ENTROPY WE CAN DETERMINE THE NONEQUILIBRIUM ENTROPY TO ONLY BY MEASURING THE HEAT CURRENTS! O(ɛ 2 ) 0 θ(t) dt + O(ɛ 3 )
OPERATIONAL DETERMINATION OF ENTROPY SUPPOSE THAT WE WANT TO DETERMINE THE DIFFERENCE S(T 1,T 2 ; ν ) S(T 1,T 2 ; ν) TEMPERATURE OF THE RIGHT BATH T 2 ONLY THE PARAMETER CHANGES ν ν = O(1) T 1 ν ν PARAMETER
DIRECT PATH FIX THE TEMPERATURES AND CHANGE S(T 1,T 2 ; ν ) S(T 1,T 2 ; ν) = T 0 ν TO ν FROM THE EXTENDED CLAUSIUS RELATION, ONE GETS θ(t) dt + O(ɛ 2 ) WE CAN DETERMINE THE DIFFERENCE ONLY WITH THE PRECISION OF O(ɛ) T 2 T 1 ν ν PARAMETER
INDIRECT PATH USE THE COMBINATION OF THE THREE PROCESSES a b (T 1,T 2 ; ν) (T (T 1,T 1 ; ν c 1,T 1 ; ν) ) (T 1,T 2 ; ν ) TEMPERATURE OF THE RIGHT BATH T 2 T 1 T 1 EQUILIBRIUM ν ν PARAMETER
INDIRECT PATH USE THE COMBINATION OF THE THREE PROCESSES (T 1,T 2 ; ν) a b (T (T 1,T 1 ; ν c 1,T 1 ; ν) ) (T 1,T 2 ; ν ) FROM THE EXTENDED CLAUSIUS RELATION, ONE GETS S(T 1,T 2 ; ν ) S(T 1,T 2 ; ν) = T 0 θ(t) a dt + Q b T 1 T 0 θ(t) c dt + O(ɛ 3 ) WE CAN DETERMINE O(ɛ 2 THE DIFFERENCE WITH THE PRECISION OF )
POSSIBLE ERROR IN EACH PROCESS TEMPERATURE OF THE RIGHT BATH O(ɛ 2 ) T 2 T 1 O(ɛ 3 ) 0 O(ɛ 3 ) EQUILIBRIUM T 1 ν ν THE TEMPERATURE OF THE LEFT BATH IS FIXED AT PARAMETER T 1
NONLINEAR NONEQUILIBRIUM RELATIONS
REPRESENTATIONS OF NESS LINEAR RESPONSE [ ρ ss (Γ) = exp C + T dt 0 j=1,2 J j (t) T j Γ eq + O(ɛ 2 ) ] KOMATSU-NAKAGAWA REPRESENTATION [ { ρ ss (Γ) = exp C + 1 T dt 2 0 j=1,2 J j (t) T j Γ ss T dt 0 j=1,2 } J j (t) T j ss Γ + O(ɛ 3 ) ] IMPROVED RELATION WHICH USES PATH AVERAGES WITH INITIAL CONDITION AND FINAL CONDITION
THE SECOND ORDER EXTENDED CLAUSIUS RELATION STARTING FROM THE KOMATSU AND NAKAGAWA S REPRESENTATION (INSTEAD OF THE LINEAR RESPONSE REPRESENTATION), WE CAN SHOW S(T 1,T 2 ; ν ) S(T 1,T 2 ; ν) = T 0 θ(t) dt + 1 W WORK DONE TO THE SYSTEM REFERENCE TEMPERATURE T Ψ = T 0 dt j=1,2 2T W ; Ψ + O(ɛ3 ) ( 1 T j (t) 1 T ) J j (t)
THE SECOND ORDER EXTENDED CLAUSIUS RELATION S(T 1,T 2 ; ν ) S(T 1,T 2 ; ν) = T 0 θ(t) dt + 1 2T W ; Ψ + O(ɛ3 ) THE RELATION TAKES INTO ACCOUNT NONLINEAR NONEQUILIBRIUM CONTRIBUTIONS, AND HAS A DESIRED HIGHER PRECISION. BUT IT CONTAINS A CORRELATION BETWEEN THE HEAT AND WORK. IT IS A RELATION BETWEEN MACROSCOPIC QUANTITIES; BUT IS IT A THERMODYNAMIC RELATION?
2nd TWIST OPERATIONAL DETERMINATION OF ENTROPY TO DETERMINE THE ENTROPY DIFFERENCE TO THE ORDER S T 0 θ(t) dt + 1 2T W ; Ψ O(ɛ 2 ) S T 2 T 0 T 1 θ(t) dt S T 0 θ(t) dt EQUILIBRIUM T 1 S = Q T ν ν ONE HAS TO USE EITHER THE 1ST ORDER RELATIONS OR THE 2ND ORDER RELATION, DEPENDING ON THE PATHS.
FULL ORDER RELATION THERE IS A JARZYNSKI-LIKE RELATION, WHICH IS VALID TO THE FULL ORDER S(T 1,T 2; ν ) S(T 1,T 2 ; ν) = 1 T {U(T 1,T 2 ; ν) U(T 1,T 2; ν )} + log exp[ (Ψ + W/T )/2] exp[ (Ψ + W /T )/2] IT IS A RELATION BETWEEN MACROSCOPIC QUANTITIES; BUT IS IT A THERMODYNAMIC RELATION??? S(T 1,T 2 ; ν) =S sym [ρ (T 1,T 2 ;ν) ss ( )] + O(ɛ 3 )
SUMMARY
WHAT DID WE GET? WE FOUND A NATURAL EXTENSION OF CLAUSIUS RELATION FOR NESS (WHEN THE DEGREE OF NONEQUILIBRIUM IS SMALL) IT ENABLES ONE TO OPERATIONALLY DETERMINE NONEQULIBRIUM ENTROPY TO THE SECOND ORDER IN ɛ = O(T 2 T 1 ) WE FOUND THAT THE CORRESPONDING ENTROPY HAS A MICROSCOPIC REPRESENTATION IN THERMS OF SYMMETRIZED SHANNON ENTROPY (THE 1st TWIST )
WHAT DID WE GET? WE FURTHER DERIVED NONLINEAR NONEQUILIBRIUM VERSIONS OF THE THERMODYNAMIC RELATIONS TO DETERMINE THE ENTROPY TO THE ORDER OF O(ɛ 2 ) WE HAVE TO USE THE 1st ORDER OR THE 2nd ORDER RELATIONS DEPENDING ON THE PATHS (THE 2nd TWIST )
FUTURE PROBLEMS OTHER THERMODYNAMIC RELATIONS? (INCLUDING THE SECOND LAW) 1st TWIST: DOES THE SYMMETRIZED SHANNON ENTROPY HAVE ANY KEYS TO STATISTICAL MECHANICS FOR NESS?? 2nd TWIST: DOES THE UNBALANCE OF THE 1st AND THE 2nd ORDER RELATIONS IMPLY ANYTHING ESSENTIAL ABOUT THE PHYSICS OF NESS??