Connec&ng thermodynamics to sta&s&cal mechanics for strong system-environment coupling

Transcript

1 Connec&ng thermodynamics to sta&s&cal mechanics for strong system-environment coupling Chris Jarzynski Ins&tute for Physical Science and Technology Department of Chemistry & Biochemistry Department of Physics University of Maryland, College Park

2 Macroscopic and microscopic machines steam engine ATP synthase RCSB Protein Data Bank

3 air (T) rubber band length = L water (T,P) μ- pipeoe RNA λ laser trap U S+E =U sys +U env U S+E = u sys +U env + u int neglect interac&on energy can t neglect interac&on energy!! heat, work, entropy 1 st & 2 nd Laws of Thermodynamics!?? Goal: microscopic defini&ons that preserve the structure of thermodynamic laws

4 Macroscopic thermodynamics air (T) rubber band length = L variables: (L,T) uilibrium state state func&ons: U internal energy S entropy Ψ tension thermodynamic poten&al: F = U ST Helmholtz free energy First Law: ΔU = Q + W W = Ψ dl Second Law: Q TΔS W ΔF Clausius inuality all about uilibrium states and thermodynamic processes

5 Sta&s&cal mechanics x = (q,p) = microstate of system U sys (x;λ) = Hamiltonian (energy func&on) state func&ons U = S = p uilibrium states (λ,t) sta&s&cal ensembles ( x;λ,t ) = 1 Z exp βu sys ( x;λ) β = 1 p U sys p ln p F = k B T ln Z fluctua&ng observable F =U ST k B T

6 Sta&s&cal mechanics x = (q,p) = microstate of system U sys (x;λ) = Hamiltonian (energy func&on) thermodynamic processes λ : A B, 0 t τ λ(t) = protocol - how we act on the system x(t) = trajectory - how the system responds τ U sys ( x τ ;B) U sys ( x 0 ; A) = dt!x U sys τ + dt! λ U sys = Q +W 0 0 x λ Average over realiza&ons: A B ΔU = Q + W W ΔF F =U ST First Law Second Law

7 x = microstate of system y = microstate of environment λ = work parameter Nanoscale sta&s&cal mechanics water (T,P) μ- pipeoe RNA λ laser trap Fluctua7on Theorems ρ R (-w) e βw = e βδf forward process: A -> B reverse process: B -> A ρ F (+w) ρ R ( w) = exp β w Δf ρ F (w) & others Δf w

8 x = microstate of system y = microstate of environment λ = work parameter Nanoscale sta&s&cal mechanics water (T,P) μ- pipeoe RNA λ laser trap Fluctua7on Theorems e βw = e βδf ρ F (+w) ρ R ( w) = exp β w Δf Δf or Δg? & others

9 Thermodynamics Sta&s&cal Mechanics uilibrium states thermodynamic processes ensembles trajectories p e βu sys F = U - ST Q + W = ΔU W ΔF F = U - ST <Q> + <W> = ΔU <W> ΔF + fluctua7on theorems U S+E ( x, y;λ) = u sys ( x;λ) +U env ( y) + u int ( x, y) can t be neglected!

10 Thermodynamics Sta&s&cal Mechanics Develop a sta&s&cal mechanical framework that: - [correctly describes the uilibrium state of a nanoscale system] - provides precise microscopic defini&on of relevant state func&ons (internal energy, entropy, free energy, etc.), heat & work - includes statements of first & second laws, fluctua&on theorems - scales up to macroscopic thermodynamics, when the system is large U S+E ( x, y;λ) = u sys ( x;λ) +U env ( y) + u int ( x, y) can t be neglected!

11 Macroscopic thermodynamics revisited air (T,P) rubber band length = L variables: (L,T,P) state func&ons: U S Ψ V internal energy entropy tension volume thermodynamic poten&als: F = U ST Helmholtz free energy H = U + PV enthalpy G = H ST Gibbs free energy First Law: ΔU = Q + W - PΔV W = Ψ dl ΔH = Q + W non-pdv work Second Law: Q TΔS W ΔG Clausius inuality

12 Nanoscale sta&s&cal mechanics - setup x = microstate of system y = microstate of environment λ = work parameter U S+E water (T,P) μ- pipeoe RNA λ laser trap ( x, y;λ) = u sys ( x;λ) +U env ( y) + u int ( x, y) fluctua&ng observable sys env (T,P) p ( x, y;λ ) = 1 exp( βu S+E ) Z S+E Define state func&ons u, s, v, h, g that describe the thermodynamic state of the system of interest.

13 p p Equilibrium state of the system of interest ( x, y;λ ) = 1 exp( βu S+E ) Z S+E ( x;λ ) = dy p ( x, y;λ ) = 1 exp( βu sys ) dy Z S+E exp β U env + u int U S+E = u sys +U env + u int e β ( u sys+φ ) φ 0 as u int 0 = 1 β ln dy exp β ( U env + u int ) dy exp( βu env ) φ x

14 Equilibrium state of the system of interest φ x p ( x;λ ) exp β ( u sys +φ) =? exp βu sys = 1 β ln dy exp β ( U env + u int ) dy exp( βu env ) = solva7on free energy for system in frozen microstate x = reversible work ruired to insert system into environment J.G. Kirkwood, J Chem Phys 3, 300 (1935) U. Seifert, Phys Rev LeO 116, (2016)

15 Solva&on free energy and volume Macroscopic system: pebble H 2 O H 2 O ϕ = solva&on free energy = reversible work of inser&on = P V peb V peb = φ / P thermodynamic defini&on of V peb Use this rela&on to define the volume of our microscopic system: v sys ( x) φ ( x) / P

16 Fluctua&ng observables and state func&ons p ( x;λ ) exp# $ β u sys +φ % & = exp β u sys + P v sys exp βh sys v sys ( x) φ ( x) / P fluctua&ng observables Now define state func&ons (of λ,t,p): u v h s dx p u sys ( x;λ) dx p v sys x dx p h sys x;λ dx p ln p g 1 β ln dx exp # $ β u + P v sys sys h = u + Pv g = h st % &

17 First Law of Thermodynamics, ΔU = Q PΔV + W u sys ( x;λ) x( t) λ h sys λ : A B ( x;λ) u sys ( x;λ) + Pv sys ( x) Δu sys = # dt!x u sys x +! λ u & sys % ( = dt!x h sys $ λ ' x P dv sys + dt! λ q heat (Q) dt!x h sys w dt! λ u sys x λ PdV work (-P ΔV) u sys λ non-pdv work (W)

18 First Law of Thermodynamics, ΔU = Q PΔV + W u sys ( x;λ) x( t) λ h sys λ : A B ( x;λ) = u sys ( x;λ) + Pv sys ( x) Δu sys = # dt!x u sys x +! λ u sys % $ λ & ( ' P dv sys + dt! λ = dt!x h sys x = q PΔv sys + w u sys λ q dt!x h sys x w dt! λ u sys λ = dt! λ h sys λ

19 First Law of Thermodynamics, ΔU = Q PΔV + W u sys ( x;λ) x( t) h sys λ : A B ( x;λ) = u sys ( x;λ) + Pv sys ( x) λ Δu sys = q PΔv sys + w for one realiza&on of the process Averaging over an ensemble of realiza&ons from one uilibrium state to another gives: Δu = q PΔv + w Δh = q + w q dt!x h sys x w dt! λ u sys λ = dt! λ h sys λ

20 Second Law of Thermodynamics, W ΔG u sys ( x;λ) x( t) λ h sys λ : A B ( x;λ) = u sys ( x;λ) + Pv sys ( x) e βw = e βδg w Δg w dt! λ u sys λ ρ F (+w) ρ R ( w) = eβ w Δg g 1 β ln dx exp β u + P v sys sys = u + Pv st

21 Summary to this point p ( x;λ ) = 1 Z exp β u sys +φ Internal energy u sys ( x;λ) Volume Enthalpy Entropy Gibbs free energy v sys ( x) = φ / P h sys = u sys + Pv sys u = u sys v = v sys h = h sys s = ln p g = β 1 ln Z Heat, Work!q =!x x h sys,!w =! λ λ u sys h = u + Pv g = h st Δh = q + w w Δg & e βw = e βδg ρ F (+w) ρ R ( w) = eβ w Δg

22 Summary to this point p ( x;λ ) = 1 Z exp β u sys +φ Internal energy u sys ( x;λ) Volume Enthalpy Entropy Gibbs free energy v sys ( x) = φ / P h sys = u sys + Pv sys u = u sys v = v sys h = h sys s = ln p g = β 1 ln Z Heat, Work!q =!x x h sys,!w =! λ λ u sys Also: u,v,h,s,g -> U,V,H,S,G for a macroscopic system of interest. bare representa&on C.J., Phys Rev X 7, (2017)

23 v sys Internal energy Volume Enthalpy Entropy Gibbs free energy Alterna&ve framework ( x) = φ / P φ T T φ T, φ P φ P P u = u + φ φ T v = v sys h = h φ T s = s φ T T g = g φ P ~ Seifert PRL (2016) Heat, Work q = q Δφ T, w = w Also: u,v,h,s,g -> U,V,H,S,G for a macroscopic system of interest. par7al molar representa&on C.J., Phys Rev X 7, (2017)

24 Deriva&on of e βw = e βδg p U S+E ( x, y;λ ) = 1 exp( βu S+E ) Z S+E ( ζ;λ) = u sys ( x;λ) +U env ( y) + u int ( x, y) ζ=(x,y) includes the piston Assume: (1) Ini&al condi&ons are sampled from p (x,y;λ). (2) Evolu&on is governed by Hamiltonian dynamics, with externally imposed protocol λ(t).

25 Deriva&on of e βw = e βδg p U S+E ( x, y;λ ) = 1 exp( βu S+E ) Z S+E ( ζ;λ) = u sys ( x;λ) +U env ( y) + u int ( x, y) λ=a trajectory λ=b ζ 0 ζ τ = ζ τ (ζ 0 ) w = τ 0 dt! λ u sys λ = τ 0 dt! U λ S+E λ =U S+E ( ζ τ ;B) U S+E ( ζ 0 ; A)

26 Deriva&on of e βw = e βδg p ( x, y;λ ) = 1 exp( βu S+E ) Z S+E w =U S+E ζ τ ;B U S+E ( ζ 0 ; A) Z S+E ( λ) = dζ exp( βu S+E ) = dx exp β ( u sys +φ) dy exp βu env = exp βg λ Z env

27 e βw = dζ 0 = λ=a Z S+E 1 ( A) Deriva&on of e βw = e βδg p w =U S+E ζ τ ;B Z S+E trajectory exp βu S+E ζ 0 ; A A Z S+E ( x, y;λ ) = 1 exp( βu S+E ) Z S+E U S+E ( ζ 0 ; A) ( λ) = exp βg( λ) Z env exp( βw) dζ 0 exp βu S+E ( ζ τ ;B) = Z B S+E A Z S+E λ=b ζ 0 ζ τ = ζ τ (ζ 0 ) = e β Δg

28 Conjugate (forward & reverse) processes Forward process: A->B Reverse process: A<-B p λ F R t = λ τ t γ F x (0) F x ( τ ) F q γ R x ( τ ) R x (0) R P F [x F t x F 0 ) P R [x R t x R 0 ) = exp βq x t ( F ) compare w/ G.E. Crooks, J Stat Phys 90, 1481 (1998)

29 Summary C.J., Phys Rev X 7, (2017) state func&ons U, V, S thermodynamic poten&als H=U+PV, G=H-ST 1 st Law: ΔH = Q+W 2 nd Law: W ΔG?

30 Summary C.J., Phys Rev X 7, (2017) p ( x;λ ) = 1 Z exp β u sys +φ Internal energy u sys ( x;λ) Volume Enthalpy Entropy Gibbs free energy v sys ( x) = φ / P h sys = u sys + Pv sys u = u sys v = v sys h = h sys s = ln p g = β 1 ln Z Heat, Work!q =!x x h sys,!w =! λ λ u sys h = u + Pv g = h st Δh = q + w w Δg & e βw = e βδg ρ F (+w) ρ R ( w) = eβ w Δg