new nonequilibrium thermodynamics at the...
TRANSCRIPT
![Page 1: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/1.jpg)
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
![Page 2: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/2.jpg)
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)
![Page 3: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/3.jpg)
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
![Page 4: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/4.jpg)
Distribution of entropy generated
!
"# S( )
0
!
S
![Page 5: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/5.jpg)
Fluctuation Theorem
!
"# S( )
0
!
S
!
"# $S( )
Distribution of entropy generated
![Page 6: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/6.jpg)
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
![Page 7: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/7.jpg)
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( )
![Page 8: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/8.jpg)
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)
![Page 9: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/9.jpg)
Demonstration of (integrated) Fluctuation Theorem
!
"# +S( )"# $S( )
= exp S /kB( )
!
˙ S
!
t
!
"
!
P S" < 0( )P S" > 0( )
= exp #S" /kB( )S" >0
Wang et al, Phys Rev Lett (2002)
l.h.s.
r.h.s.Ideally: divide the trajectory into many segments of duration τ, compute S for each segment, & construct histogram … not enough data !
![Page 10: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/10.jpg)
More recently …
!
V z;I( ) = A " exp(#$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)
![Page 11: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/11.jpg)
displacement (λ)
isothermal& reversible
irreversible
!
W = f d"A
B
#$Wrev
= %F = FB & FA
A
B
Illustration of Clausius inequality: stretched rubber band
rubber band
spring
particlewall
λ
air
force (f)
![Page 12: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/12.jpg)
Single-molecule pullingexperiment
RNA strand
bead bead
!
H2O
micro-pipette
lasertrapDNA handles
!
"
1. Begin in equilibrium2. Stretch the molecule W = work performed3. End in equilibrium4. Repeat
λ = Α λ : Α Β
λ = Β
Irreversible process:
![Page 13: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/13.jpg)
After infinitely many repetitions …
“violations”of 2ndLaw
distribution ofwork values
!
"F = FB# F
A
average workperformed
![Page 14: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/14.jpg)
Nonequilibrium work theorem
!
e"#W
= e"#$F
!
dW" #(W )e$%W
!
1/kBT
• valid far from equilibrium• fluctuations in W satisfy strong constraint• nonequilibrium measurements reveal equilibrium properties
C.J., Phys Rev Lett (1997)Crooks, J Stat Phys (1998)Hummer & Szabo, PNAS (2001) & others
![Page 15: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/15.jpg)
Derivation for isolated system,p. 1
!
H(x;")microstateHamiltonian !
"
Derivation (isolated system)
!
x positions, momentainternal energy
In equilibrium …
!
peq(x;") =
1
Z"
e#$H (x;") Boltzmann-Gibbs distribution
!
Z" = dx# e$%H (x;" ) partition function
!
F" = #$#1lnZ"
free energy
![Page 16: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/16.jpg)
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
![Page 17: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/17.jpg)
!
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"#$F QED
λ=A λ=Btrajectory
!
x0
!
x" = x" (x0)
Derivation for isolated system,p. 3
![Page 18: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/18.jpg)
Various derivations• C.J. PRL & PRE 1997, J.Stat.Mech. 2004• G.E. Crooks J.Stat.Phys. 1998, PRE 1999, 2000• G. Hummer & A. Szabo PNAS 2001• S.X. Sun J.Chem.Phys. 2003• D. J. Evans Mol.Phys. 2003• S. Mukamel PRL 2003 … & others
Hamiltonian evolution, Markov processes,Langevin dynamics, deterministic thermostats,quantum dynamics … robust
related result: G.N. Bochkov & Y.E. Kuzovlev JETP 1977
see also S. Park & K. Schulten, J. Chem. Phys. 2004 R.C. Lua & A.Y. Grosberg, J. Phys. Chem. B 2005
![Page 19: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/19.jpg)
Experimental verification: unfolding a single RNA molecule
Liphardt et al, Science (2002)
unfolding / refolding cycles
slow
fast
hysteresis(irreversibility)
![Page 20: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/20.jpg)
Results: equilibrium ΔF from nonequilibrium work values
• three pulling rates: 2-5 pN/s , 34 pN/s , 52 pN/s• ~ 300 cycles at each rate• slow cycles (reversible) used to determine ΔF
Ener
gy (k
T)
Extension (nm)5 10 15 20 25 30
J.Liphardt et al, Science (2002)
!
W "#F
!
W "#
2$W
2 "%F
!
"#"1ln e
"#W "$F
![Page 21: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/21.jpg)
Relation to Second Law
!
P[W < "F #$ ] = 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
![Page 22: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/22.jpg)
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 $&F( )[ ]
Crooks, Phys Rev E (1999)
Experimental verification:Collin et al, Nature (2005)
![Page 23: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/23.jpg)
3-helix junction force-extension curves
3-helix junction of ribosomal RNA of E. coli
!
Wdiss
" 50kBT
wild type mutant
Collin et al, Nature 2005
![Page 24: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/24.jpg)
Verification of CFT!
"A#B
(+W )
"B#A
($W )= exp % W $&F( )[ ]
!
ln"A#B
(+W )
"B#A
($W )= %(W $&F)
![Page 25: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/25.jpg)
Macroscopic machines
steam engine Carnot cycle
textbook thermodynamics
![Page 26: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/26.jpg)
Molecular machines
RNA polymerase
ATP synthasekinesin
What are the underlying thermodynamics?How is chemical energy converted to mechanical motion?
![Page 27: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/27.jpg)
Simple explanation ofmolecular machines
ADP
ATP
ADP
ATP track
currents:
!
˙ x
!
˙ "
stepping velocity
ATP -> ADP conversion rate
!
˙ " = #" /$
!
˙ 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.
![Page 28: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/28.jpg)
Minimal molecular motors Van den Broeck, Meurs, Kawai Phys Rev Lett (2004)
T2T2
T1 T1
“Triangula” “Triangulita”
Directed motion?
YES !
![Page 29: New Nonequilibrium thermodynamics at the microscaleonline.itp.ucsb.edu/online/biomachine06/jarzynski/pdf/... · 2006. 5. 26. · Nonequilibrium thermodynamics at the microscale C](https://reader033.vdocument.in/reader033/viewer/2022060711/60770f753369d13f85533933/html5/thumbnails/29.jpg)
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