Download Entropy production and fluctuation relations from full phase space

Entropy production and fluctuation
relations from full phase space
stochastic dynamics
Ian Ford and Richard Spinney
Department of Physics and Astronomy and
London Centre for Nanotechnology
University College London, UK
http://arxiv.org/pdf/1201.0904v2.pdf
http://arxiv.org/pdf/1203.0485v1.pdf
Summary
• Review of stochastic thermodynamics
– extended to full phase space (positions+velocities)
• Separation of entropy production into three
components, including a new one: S3.
– statistical properties of each
• Applications: entropy production due to
– particle hopping in discrete phase space
– thermal conduction in continuous phase space
Non-equilibrium thermodynamics of entropy
production
S tot  Ssys  S med
Ssys    P ln Pdx   ln P
S med 
Qmed
Tmed

Qhk
Tmed

medium
system
Qexcess
Tmed
House-keeping heat transfer to the medium is the
entropy production in a non-equilibrium stationary state:
Oono and Paniconi (1998)
Non-equilibrium thermodynamics of entropy
production
S tot  Ssys  S med
medium
path averaging
Ssys    P ln Pdx    ln P
S med 
Qmed
Tmed

Qhk
Tmed

system
Qexcess
Tmed
Is there a microscopic entropy change that when pathaveraged becomes the thermodynamic entropy change?
Stochastic thermodynamics
• Stochastic motion of an open system
– by master equation or SDE
Seifert,
Sekimoto,
Kurchan,
Lebowitz,
Spohn, etc
• Stochastic evolution of the total microscopic entropy
entropy
position
time
Microscopic entropy and its evolution
• Let’s consider a measure of irreversibility

x
position
position
– the relative likelihood of observing a reversal of behaviour
time
Under forward dynamics
R
x
time
Under reversed dynamics
Entropy production 2
• Define total entropy change
S tot

 ln 


F

T
( x f | xi ) P( xi ,0) 
p( x ) 

  ln  R
R R 
 T ( x | x ) P( x , t ) 
p (x ) 
i
f
f


initial
xf
xi
0
t
P
final
pdf
position
Conditional path probabilities
xi
time
xf
position
Entropy production 3
S tot
 T F ( x f | xi ) 
 P( x f , t ) 

  ln  R
  ln 
 T (x | x ) 
P
(
x
,
0
)
i
i
f 



 T F ( x f | xi ) 

  ln P( x f , t )  ln P( xi ,0)   ln  R
 T (x | x ) 
i
f 

 Ssys  S med
• The path-average of the last term corresponds to
the medium entropy change of Oono and Paniconi
– (house-keeping and excess heat transfers divided by
medium temperature).
Integral fluctuation relation
• The total entropy change satisfies an integral
fluctuation relation
For any two dynamical
schemes that generate paths
in a 1:1 correspondence
Non-negative average microscopic entropy
change
S tot

 T F ( x f | xi ) P( xi ,0) 
 p( x ) 


 ln R
 ln  R  R 
 T ( x | x ) P( x , t ) 
 p (x ) 
i
f
f


so
S tot  0
• But what about a reversal of a path in full phase
space?
– coordinates that change sign upon time-reversal?
t
Generalisation of total entropy production for
odd variable dynamics
S tot
 T F (v f | vi ) P(vi ,0) 
 T F (v f | vi ) P(vi ,0) 
  ln 

 ln  R
T
R
 T ( v | v ) P ( v , t ) 
 T ( v | v ) P (v , t ) 
i
f
f
i
f
f




• Reversal of the forward path is the
– inverted reverse velocity sequence
– starting from the inverted distribution PT
– of the inverted final velocity from the forward path
The three components of entropy production
• Total entropy change is now defined as
velocity-inverted
reversed path
• Identify two more quantities that satisfy an integral
fluctuation relation and which contribute to Stot
• We consider two different possibilities for the
-dynamics and -path
Forward path under normal
dynamics
V  2
x
V  1
V  1
V  2
x†
X 1
2
3
4
Velocity-inverted reversed path
under time-reversed normal
dynamics
Reverse path under timereversed adjoint dynamics
Forward path under normal
dynamics
V  2
x
xR
V  1
V  1
V  2
x†
X 1
2
3
4
Velocity-inverted reversed path
under time-reversed normal
dynamics
X 1
2
3
4
Reverse path under timereversed adjoint dynamics
Forward path under normal
dynamics
V  2
x
xR
V  1
V  1
V  2
x†
X 1
2
3
4
Velocity-inverted reversed path
under time-reversed normal
dynamics
xT
X 1
2
3
4
Velocity-inverted forward path
under adjoint dynamics
Two entropy change-like quantities
• S1 and S2 satisfy integral fluctuation relations,
hence are positive in the mean
• Furthermore
S tot  S1  S 2  S3
• These map onto
Qexcess Qhk
S tot  Ssys 

T
T
med
med
Following Esposito and van den Broeck (2010),
but third term is new
The properties of S1,2,3 are various:
• S1 vanishes in a stationary state
– entropy production associated with relaxation
• S2 and S3 vanish for detailed balance
– entropy production associated with house-keeping heat
• S3 vanishes in the mean in a stationary state
– it is associated with relaxation as well
• S3 vanishes in the absence of odd variables
– or if the stationary state is symmetric in odd variables
• The sign of the average of S3 is unbounded
– in contrast to that of S1 , S2 , and Stot : never negative
Simple example: driven and diffusive particle
motion on a ring
velocity


X1
X2

position
Stochastic dynamics generates a path
consisting of residence and transitions

XL
Master equation transition rates
X1+
X2+
T ( X i 1  | X i )  C
T ( X i  | X i )  A
T ( X i  | X i )  B
T ( X i  | X i 1 )  C
X1
X2
...and entropy production
T ( X i  | X i )   B  C  S3  0 S2  ( A  B)t
X1+
X2+
T ( X i 1  | X i )  C
T ( X i  | X i )  B
 S3  2 ln( A / B)
S 2   ln( A / B)
 S3  0 S 2  0
T ( X i  | X i )  A
T ( X i  | X i 1 )  C
S 2  ln( A / B)
 S3  2 ln( A / B)
 S3  0 S 2  0
X1
X2
T ( X i  | X i )   A  C  S3  0 S2  ( A  B)t
A string of entropy increments
velocity


X1
X2
X3
X4


XL
position
Stochastic dynamics generates a path
consisting of residence and transitions
S 2  S 2 ( X 2  X 3 )  S 2 ( X 3 , t )  S 2 ( X 3  X 4 )
 S 2 ( X 4 , t )  S 2 ( X 4  X 4 )  S 2 ( X 4 , t )
Starting to resemble:
entropy
position
time
For this simple model components of the mean
house-keeping entropy production rate are
t
0
t
t
t
In the stationary state:
d S 2
dt
stat
( A  B) 2

( A  B)
d S3
dt
stat
0
Division of the house-keeping heat of Oono
and Paniconi
• Generalised house-keeping heat
Qhk,G  S 2Tmed
• Transient house-keeping heat
Qhk,T  S3Tmed
On into the continuum
• SDEs for positions, velocities and the components
of entropy production..
dS tot   dxi   dt
and
dS1,2,3  
Thermal conduction: Brownian particle
in a trap with a temperature gradient
temperature
Trap potential
position
Thermal conduction: Brownian particle
in a trap with a temperature gradient
temperature
Trap potential
position
Thermal conduction: Brownian particle
in a trap with a temperature gradient
temperature
Trap potential
position
Thermal conduction: Brownian particle
in a trap with a temperature gradient 2
• Essential to include velocity variable: try an
overdamped treatment:
dS tot
0
??
Thermal conduction: Brownian particle
in a trap with a temperature gradient 3
• employ full phase space dynamics:
dS tot  dS sys
 dEsys  dW  / T ( x)  dS med
Distributions of total entropy production in
stationary state of thermal conduction
Increasing
time
interval
Stot also satisfies a detailed fluctuation
relation
p (S tot )
 exp( S tot )
p ( S tot )
Mean rate of total entropy production in a
stationary state of thermal conduction
Leading order term in 1/
According to classical thermodynamics
Distributions of entropy production S3
associated with transient house-keeping heat
Increasing
time
interval
Full phase space treatments are vital for a
proper description of entropy production in:
•
•
•
•
Particle diffusion down chemical potential gradient
Evaporation/condensation at surfaces
Particle escape from a potential well
Dynamics of spinning objects, magnetic moments
• In each case S3 exists, contributes to the housekeeping heat, and has
– zero mean in stationary state
– positive or negative mean in approach to stationary state
Conclusions
???entropy???
• A microscopic measure of irreversibility in stochastic systems
exists and when path-averaged equals thermodynamic entropy
production (Seifert, Sekimoto etc).
• To understand entropy production in general cases we must
consider dynamics in full phase space.
• Entropy production separates into three components: two of its
components and itself satisfy integral fluctuation relations and are
thus never negative on average (Esposito and van den Broeck).
• The third component S3 is new and arises when the stationary
state is asymmetric in odd variables: its average can take either
sign during relaxation, but is zero in a stationary state.
• We develop entropy production with full phase space dynamics
and demonstrate the properties of S3 : for example to treat the
phenomenon of thermal conduction.