Fluctuation Theorem & Jarzynski Equality Zhanchun Tu ( 涂展春 ) Department of Physics Tamkang University — new developments in nonequilibrium process

Outline I. Introduction II. Fluctuation theorem (FT) & Jarzynski equality (JE) III. Generalized JE IV. Proposed experiments & Summary

I. Introduction

Thermodynamics Object (for classic thermodynamics) –Systems: a large number of particles (~10 23 ) –Short-range interaction between particles (ideal gas, vdW gas, plasma, polymer; gravity system, + or - charged system) –An isolated system can reach thermal equilibrium through finite-time relaxation

Four thermodynamic laws –0th: it is possible to build a thermometer –1st: energy is conserved –2nd: not all heat can be converted into work –3rd: absolute zero temperature cannot be reached via a finite reversible steps 1st+2nd+const. T:

Statistical mechanics Function Time reversible Macroscopic, reversible Time reversible Time irreversible ? Newtonian mechanics N-particle system Thermodynamics Stat. Mech. (Ensemble average)

Macrostate: thermodynamic EQ state –e.g. PVT, HMT etc. Microstate: phase point (q,p) A macrostate q p Each macrostate corresponds to many microstates!

Non-equilibrium process The system is driven out of the equilibrium by the external field Two equalities are proved to hold still for the system far away from the equilibrium FT: probability of violating the 2nd Law of thermodynamics along a micropath in NEQ process JE: extract free energy difference between two EQ states from the NEQ work performed on the system in the process between these two EQ states Classic system external field (finite time interval)

II. FT & JE

FT [Adv. Phys. 51 (2002) 1529] q p B t=t 2 q p A γ(t) t=t 1 Entropy production function Phase space contraction factor FT: p(s) represents the probability distribution of the entropy production function taking the value s along the micro-path γ

Preconditions for FT –Microscopic dynamics: time reversible –Initial distribution is symmetric under the time reversal mapping –Ergodic consistency: Time reversal mapping Γ m (0) Γ m (t)

JE [PRL 78 (1997) 2690] JE: w : work performed on the system along each micro-path ΔF : free energy difference between two macrostates : average for all micro-paths t=t 2 t=t 1 Macrostate 1 Temperature T Macrostate 2 Temperature T q p q p B1B1 A1A1 B2B2 A2A2 B3B3 A3A3 … γ1γ1 γ2γ2 γ3γ3

Examples for JE Gas & piston Unfolding RNA hairpin [Nature 437 (2005) 231]

Relation between FT & JE FT  JE –Crooks: proof for stochastic, microscopically time reversible dynamics [PRE 60 (1999) 2721] –Evans: proof for time reversible deterministic dynamics [Mol. Phys. 101 (2003) 1551]

q p Evans’ proof: FT  JE General description q p B t=t 2 q p γ(t) t=t 1 t=t’ 2 Macrostate 1 Temperature T Macrostate 2 Temperature T EQ Switch external field (parameter from λ 1 to λ 2 ) Long time relaxation Attention: the system always contact with the constant temperature thermal bath Do work! No work A

FT between 2 macrostates –Note: Original FT is valid for the ensemble containing all paths beginning from all microstates at time t 1 –Evans: FT holds also for the ensemble only containing all paths connecting the microstates corresponding to macrostates 1 and 2 [Mol. Phys. 101 (2003) 1551] <>: average for all micro-paths beginning from the microstates corresponding to the macrostate 1

Initial and final distribution functions Phase space contraction factor Effective dynamics of isothermal system Number of particles thermostat multiplier ensuring the kinetic temperature of the system to be fixed at a temperature T. It reflects the heat exchange between the system and the thermal bath. Crucial condition in the derivation of JE from FT!

Energy conservation along the micro-path Work performed on the system along the micro-path Heat absorbed by the system form the thermal bath along the micro-path Entropy production function JE

FT, JE and 2nd law FT permits the existence of micro-pathes violating the 2nd law [Crooks FT, 1999] JE: macro-work satisfies 2nd law w1w1 w2w2 <w><w> w p(w)p(w) ΔFΔF RNA unfolding [Nature 437 (2005) 231]

Remarks –JE satisfies 2nd law –2nd law cannot be derived from JE: 2nd law holds in a much wider realm than JE does –Proof of JE implies microscopic time reversibility can result in macroscopic time irreversibility

III. Generalized JE (GJE) cond-mat/

Two gedanken expts. on JE Expt. 1 [J. Phys. Chem. B 109 (2005) 6805] v vpvp

Expt. 2 JE is violated!

Why? Jarzynski and Crooks’ argument: the JE fails because the initial distribution function is not canonical in the second expt. Our viewpoint: the initial distribution function is still canonical but a more underlying reason makes the JE fail. In other words, a generalized JE may exist.

Investigate 2 gedanken expts. Crucial condition (emphasize again) –time integral of the phase space contraction factor is exactly expressed as the entropy production resulting from the heat absorbed by the system from the thermal bath The dynamics of two gedanken expts. may not be described as the form of Hamiltonian dynamics with the thermostat multiplier, we should check whether the crucial condition holds or not for 2 gedanken expts.

Expt. 1 Effective dynamics (ideal gas) Influence of piston movement Phase space contraction factor (Crucial condition still holds) Thus JE holds! Switch parameter is V [Evans’s book]

Expt. 2 Effective dynamics (ideal gas) Phase space contraction factor Crucial condition does not hold  JE fails! Volume expansion has no effect on the momentum of the particles

Expt. 2 (continued) However, following the derivation of JE from FT, we obtain We have known: Although JE fails, but above general form holds! Hint: a more general version of JE may exist.

GJE Time integral of phase space contraction factor GJE Special cases: Conjecture: most of macroscopic systems satisfy σ=0.

Heuristic viewpoint Expt. 1: M→∞ Expt. 2: M→0 Intermediate case: 3rd gedanken expt. –The experimental setup is same as the first one. The mass of the piston M is finite. At time t 1, we remove the pins P 1 and P 2. The gas will push the piston to the right wall of the container. Once the piston contacts with the wall, it adheres to the wall without bounce. After a long time relaxation, the system arrives at an equilibrium state at time t 2.

3rd gedanken expt. Effective dynamics GJE for 3rd gedanken expt.

Determine g: numerical simulation Ideal gas: βm=1, 1000<N< <M/m<1000, 1.1<V 2 /V 1 <1.9 <>: average on 500 systems with different initial microstates corresponding to the same macrostate.

Fitting results Small x (inset): ν=0.53 (fitting)

Prediction Take M/m=4000, N=1000 and V 2 /V 1 =1.1 Numerical result: g= This fact implies that our conjecture on the form of g is reasonable!

IV. Proposed expts. & summary

Proposed expts. Inert gas in a long single-walled CNT 1.The friction between C 60 and (10,10) SWNT is very small 2.Long (10,10) SWNT can be achieved in recent nanotech 3.The inert gas with large radius, for example Ar, cannot go out from the small gap between C 60 and (10,10) SWNT. 4.The whole setup is put in vacuum 5.Small CNT coordinates the initial position of C 60. At some time, pull out the small CNT suddenly to a new position. Measure the velocity of C 60 when it arrives at the new position. 6.Repeat expt. and calculate σ. We expect σ ≠ 0.

Polymer in CNT L 1. Initial position L=L 1 2. Finial position L=L 2 3. L 1 <L 2 <R 0 Special macroscopic system

Summary FT & JE Micro. reversibility  macro. irreversibility Three gedanken expts. are analyzed, which implies a generalized JE may exist σ=0 for most of macroscopic systems Expts. for σ≠0 are proposed

Thank you for your attention!