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Nonequilibrium thermodynamicsat the microscale
C. JarzynskiT-13 Complex Systems , LANL
LAUR-05-2386
• The entropy of a closed system never decreases.• Clausius inequality• Carnot limit on the efficiency of heat engines
The entropy of a closed system never decreases.
fluid
paddles weight
potential energy of weight
thermal energy of fluid
~10 23 molecules
!
S
!
t
!
dS
dt= "mg
dz
dt#1
T> 0
Joule experiment (1845)
Scale down to the microscopic level …
fluid
paddles weight
potential energy of weight
thermal energy of fluid
~10 3 molecules
!
S
!
t
!
dS
dt> 0
Quantify the fluctuations!τ
Distribution of entropy generated
!
"# S( )
0
!
S
Fluctuation Theorem
!
"# S( )
0
!
S
!
"# $S( )
Distribution of entropy generated
Fluctuation Theorem
!
"# S( )
0
!
S
!
"# $S( )
!
"# +S( )"# $S( )
= exp S /kB( )
• very general• valid far from equilibrium• reduces to linear response near equilibrium
Evans & Searles, PRE 1994Gallavotti & Cohen PRL 1995Kurchan, 1998Lebowitz & Spohn, J Stat Phys 1999 + many others
Distribution of entropy generated
Sheared fluid
!
dS
dt= "
1
T# $VPxy t( )
shear rate pressure tensorvolume
Evans, Cohen, Morriss, Phys Rev Lett (1993)
moving wall
fixed wall
fluid
!
P xy
!
0.0
!
"1.0
!
"4.0
!
"3.0
!
"2.0
!
1.0
!
0.000
!
0.004!
0.006
!
0.002
!
p P xy( ) !
S"#P xy
!
"# +S( )"# $S( )
= exp S /kB( )
r
k ~ 0.1 pN/µm
!
Fopt= -kr
!
dS
dt=1
T
r F opt "
r v opt
vopt
!
˙ S
!
t
Optically dragged beads
focusedlaser light
bead
Wang et al, Phys Rev Lett (2002)
Demonstration of (integrated) Fluctuation Theorem
!
"# +S( )"# $S( )
= exp S /kB( )
!
˙ S
!
t
!
"
!
P S" < 0( )P S" > 0( )
= exp #S" /kB( )S" >0
Wang et al, Phys Rev Lett (2002)
l.h.s.
r.h.s.Ideally: divide the trajectory into many segments of duration τ, compute S for each segment, & construct histogram … not enough data !
More recently …
!
V z;I( ) = A " exp(#$z) + B + CI( ) " z
Parameter-dependent potential:
laser intensity
displacement of bead from wall
Drive the system away from equilibrium with a time-symmetric pulse (squeeze, then relax) … measure the work performed on the bead
Blickle et al, Phys Rev Lett (2006)
displacement (λ)
isothermal& reversible
irreversible
!
W = f d"A
B
#$Wrev
= %F = FB & FA
A
B
Illustration of Clausius inequality: stretched rubber band
rubber band
spring
particlewall
λ
air
force (f)
Single-molecule pullingexperiment
RNA strand
bead bead
!
H2O
micro-pipette
lasertrapDNA handles
!
"
1. Begin in equilibrium2. Stretch the molecule W = work performed3. End in equilibrium4. Repeat
λ = Α λ : Α Β
λ = Β
Irreversible process:
After infinitely many repetitions …
“violations”of 2ndLaw
distribution ofwork values
!
"F = FB# F
A
average workperformed
Nonequilibrium work theorem
!
e"#W
= e"#$F
!
dW" #(W )e$%W
!
1/kBT
• valid far from equilibrium• fluctuations in W satisfy strong constraint• nonequilibrium measurements reveal equilibrium properties
C.J., Phys Rev Lett (1997)Crooks, J Stat Phys (1998)Hummer & Szabo, PNAS (2001) & others
Derivation for isolated system,p. 1
!
H(x;")microstateHamiltonian !
"
Derivation (isolated system)
!
x positions, momentainternal energy
In equilibrium …
!
peq(x;") =
1
Z"
e#$H (x;") Boltzmann-Gibbs distribution
!
Z" = dx# e$%H (x;" ) partition function
!
F" = #$#1lnZ"
free energy
Derivation for isolated system,p. 2
!
"
Derivation (isolated system)
protocoltrajectory
work!
"t
!
0 " t " #
how we act on the systemhow the system responds
!
xt
One realization …
!
W = H(x" ;B) #H(x0;A)
1st law : ΔE = W + Q
… now take average of e-βW over many realizations
!
W (x0) = H(x" (x0);B) #H(x0;A)
!
e"#W
= dx0$1
ZA
e"#H (x0 ;A ) e
"#W (x0 )
!
=1
ZA
dx0" e
#$H (x% (x0 );B )
!
=1
ZA
dx"
#x"#x
0
$1
% e$&H (x" ;B )
=1 (Liouville’s thm.)
!
=ZB
ZA
= e"#$F QED
λ=A λ=Btrajectory
!
x0
!
x" = x" (x0)
Derivation for isolated system,p. 3
Various derivations• C.J. PRL & PRE 1997, J.Stat.Mech. 2004• G.E. Crooks J.Stat.Phys. 1998, PRE 1999, 2000• G. Hummer & A. Szabo PNAS 2001• S.X. Sun J.Chem.Phys. 2003• D. J. Evans Mol.Phys. 2003• S. Mukamel PRL 2003 … & others
Hamiltonian evolution, Markov processes,Langevin dynamics, deterministic thermostats,quantum dynamics … robust
related result: G.N. Bochkov & Y.E. Kuzovlev JETP 1977
see also S. Park & K. Schulten, J. Chem. Phys. 2004 R.C. Lua & A.Y. Grosberg, J. Phys. Chem. B 2005
Experimental verification: unfolding a single RNA molecule
Liphardt et al, Science (2002)
unfolding / refolding cycles
slow
fast
hysteresis(irreversibility)
Results: equilibrium ΔF from nonequilibrium work values
• three pulling rates: 2-5 pN/s , 34 pN/s , 52 pN/s• ~ 300 cycles at each rate• slow cycles (reversible) used to determine ΔF
Ener
gy (k
T)
Extension (nm)5 10 15 20 25 30
J.Liphardt et al, Science (2002)
!
W "#F
!
W "#
2$W
2 "%F
!
"#"1ln e
"#W "$F
Relation to Second Law
!
P[W < "F #$ ] = dW %(W )#&
"F#$
'
!
" dW #(W )$%
&F$'
( e) (&F$' $W )
!
" e# ($F%& ) dW '(W )%(
+(
) e%#W
= exp(%& /kT)
What is the probability thatthe 2nd law will be “violated”by at least ζ units of energy?!
e"#W
= e"#$F
!
W " #F
exponentially rare
Crooks fluctuation theorem
Forward process … λ : A -> BReverse process … λ : B -> A
(unfolding) (folding)
!
"A#B
W( )!
"B#A
$W( )
!
"A#B
(+W )
"B#A
($W )= exp % W $&F( )[ ]
Crooks, Phys Rev E (1999)
Experimental verification:Collin et al, Nature (2005)
3-helix junction force-extension curves
3-helix junction of ribosomal RNA of E. coli
!
Wdiss
" 50kBT
wild type mutant
Collin et al, Nature 2005
Verification of CFT!
"A#B
(+W )
"B#A
($W )= exp % W $&F( )[ ]
!
ln"A#B
(+W )
"B#A
($W )= %(W $&F)
Macroscopic machines
steam engine Carnot cycle
textbook thermodynamics
Molecular machines
RNA polymerase
ATP synthasekinesin
What are the underlying thermodynamics?How is chemical energy converted to mechanical motion?
Simple explanation ofmolecular machines
ADP
ATP
ADP
ATP track
currents:
!
˙ x
!
˙ "
stepping velocity
ATP -> ADP conversion rate
!
˙ " = #" /$
!
˙ x = "x /#
equilibriumfluctuations
!
p"eq +#x,+#$( ) = p"
eq %#x,%#$( )(detailed balance)
however …
!
"x # "$ % 0 (generically)
increase cATP
!
" + ˙ #
!
" + ˙ x motion!
A similar argument can be made far from equilibrium.
Minimal molecular motors Van den Broeck, Meurs, Kawai Phys Rev Lett (2004)
T2T2
T1 T1
“Triangula” “Triangulita”
Directed motion?
YES !
Summary
New & interesting thermodynamics at the microscale
• Fluctuation theorem symmetry between entropy generation & consumption
• Nonequilibrium work theorem equilibrium thermodynamic information encoded in fluctuations far from equilibrium
• Molecular motors “generic”
!
"# +S( )"# $S( )
= exp S /kB( )
!
e"#W
= e"#$F
