Beyond the Second Law of Thermodynamics

C. Van den Broeck

R. Kawai

J. M. R. Parrondo

The Second  Law of Thermodynamics  There exists no thermodynamic transformation whose sole effect is to extract a quantity of  heat from a given heat reservoir and to convert it entirely into work. William Thomson  (Lord Kelvin)

There exists no thermodynamic transformation whose sole effect is to extract a quantity of  heat from a colder reservoir and to deliver it to a hotter reservoir. Rudolf Clausius

Entropy and the Second Law

 S≥0

 S≥

Q T

Q

T

Isolated Systems

Closed Systems

No exchange of energy or matter between the system and the environment is allowed.

Energy exchange is allowed but not matter exchange.

Second Law

Time's Arrow!

Sir Arthur Eddington

“The law that entropy always increases holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equation – then so much the worse for Maxwell's equations ... but if your theory is found to be against the second law of thermodynamics, I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” (1928)

“The second law of thermodynamics is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of the basic concepts, it will never be overthrown.” (1949) Albert Einstein

Why is the second law an inequality?

Q  S− = T

Holy Grail of Statistical Mechanics

?

≥0

S=S r S i Q T

reversible entropy change

 Sr =

irreversible entropy production

 S i ≥0 Ludwig E. Boltzmann (1844-1906)

Second Law with Work

 U=W Q

(First Law of Thermodynamics)

 F =U −T  S

(Helmholtz Free Energy)

W − F =T S−Q= Holy Grail ≥0

W =W rev W dis reversible work

W rev = F

dissipative work

W dis ≥0

Holy Grail Revealed in Phase Space

F q , p ,t  〈W 〉− F =k B T ∫ F q , p ,t ln dq dp B q ,−p ,t  =k B T D F || B 

A Non-Equilibrium Process: Time-Dependent Hamiltonian

t0 equilibrium

T

H q , p ; 0 

W

t1

t H q , p ;t 

H q , p ; 1 

Forward Process

Q

Q' Backward Process H q , p ;0 

T H q , p ; 1 

H q , p ;t 

W'

equilibrium

Example T

0F W

L1

Forward

T

NonEquilibrium

T

L2 Backward

V

L

L2

− F W 0

Phase Space Trajectory and Density 6 N -dimension phase space q 1, p1 

[q t , p t ]

t1

p

q=q1 , q 2 , ⋯, q 3N  position p= p 1, p 2, ⋯, p 3N  momentum [q t , p t ]=phase trajectory

t0

q , p ,t =probability density

q 0, p0 

q Liouville Theorem

 q 0, p 0, t = q t , p t ,t = q 1, p1, t 1 

Microscopic Time Reversibility

q 0, p 0 q 1, p 1  q 1,− p1 q 0,−p 0 

Joseph Liouville (1809-1882)

Thermal Equilibrium and Gibbs Entropy Equilibrium Density

1 eq q , p = exp [− H q , p] Z Z=∫ exp[− H q , p]dq dp (partition function)

eq q , p=eq q−p  (detailed balance) H q , p=−k B T ln Z −k B T ln eq q , p Gibbs Entropy

S=−k B ∫ q , p ln  q , pdq dp S t 0 =S t =S t 1 

J. Willard Gibbs (1839-1903)

Definition of Work

q 1, p 1  W

q 0, p 0  t0

t1

W q 0, p 0 =H q 1 , p 1 ; 1 −H q 0 , p 0 ; 0  Statistical Average

〈W 〉=∫ q 0, p 0 ; t 0 W q o , p 0 dq 0 dp0 =∫ q 0 , p 0 ; t 0 [H q 1 , p 1 ; 1 −H q 0, p 0 ; 0 ]

Proof 〈W 〉=∫ q 0 , p 0 ; t 0 [H q 1 , p 1 ; 1 −H q0, p0 ; 0 ] =−kT ∫ F q 1, p1, t 1 ln B q 1,−p1, t 1 dq 1 dp 1 kT ∫ F q0, p0, t 0 ln F q 0, p0, t 0  dq 0, dp 0 kT ln Z 0 /Z 1  =−kT ∫ F q , p , t ln B q ,−p ,t  dq dp kT ∫ F q , p ,t  ln F q , p ,t dq dp  F

[

]

 F q , p , t  〈 W 〉 − F =kT ∫ F q , p ,t ln dq dp=kT D  F∥ B  B q ,−p ,t 

Relative Entropy (Kullback-Leibler distance) D ∥=∫  x ln

x  dx x 

x ≥0,  x ≥0 ;∫  x dx =∫  x dx =1 D∥ is a `distance` between two densities. D ∥≥0,

D ∥=0 iff  x = x 

exp[−D ∥] is a measure of the difficulity to statistically distinguish two densities. (Stein's lemma)

D ∥≥D  ∥  if  and  have less information than  and 

Relative Entropy: Exercise with Dice

normal

p 1=

1 6

p 2=

1 6

p 3=

biased

q 1=

1 3

q 2=

1 6

p 4=

1 12

q 3=

1 6

p 5=

1 6

p 6=

1 6

1 12

q 4=

1 12

q 5=

1 6

q 6=

1 4

6

pi D p∥q=∑ pi ln =0.163⋯ qi i =1 Find which dice you have by rolling it N times.

⋯ If you guess it is the normal one the probability that you are wrong is

P err N =e−N D  p∥q  ,

P err 10=0.196,

P err 20=0.04,

perr 50=0.00028

Relative Entropy and Reduced Information

normal dice

1 p odd =p1 p 3 p 5= , 2

7  odd =q 1q 3q 5 = , biased dice q 12

p even =p 2 p 4 p 6=

1 2

5 q even =q 2q 4 q 6 = 12

p odd p even D  p ∥q = p odd ln  p even ln =0.014 q odd q even D  p∥q D  p ∥q 

Dissipation and Time's Arrow

[

]

F q , p ,t  〈 W 〉− F =kT ∫ F q , p , t ln dq dp=kT D F∥B  B q ,−p , t 

D F∥B ≥0 If F =B ,



Second Law

D F∥B=0



No Dissipation

Dissipation is a quantitative measure of Irreversibility (time's arrow)!.

Slow Expansion Forward q 0, p 0 

Backward q 0, −p 0 

t =0

t =0

t=/2

t=/ 2

t=

t =

q 1, p1 

q 1,− p1  No Dissipation

Rapid Expansion Forward q 0, p 0 

Backward q 0,− p0 

t =0

t =0

t=/2

t=/2

t=

t=

q 1, p1 

q 1,−p 1 

Which direction is the triangle moving?

Jarzinski equality and Crooks Theorem

[

]

F q , p ,t  〈 W dis 〉 =kT ∫ F q , p ,t ln  q ,− p ,t  dq dp B Work at a phase point

F q , p ,t  W dis q , p ,t =kT ln B q ,−p , t 

Crooks theorem

(can be negative)

B q ,−p ,t  exp[− W dis q , p , t ]= F q , p ,t 

Jarzynski equality

〈 exp[−W dis ] 〉 =∫ F q , p ,t exp[− W dis q , p ,t ]dq dp=1

Coarse Graining Devide the whole phase space into N subsets  j  j =1⋯N  j

j

F t =∫ F q , p , t dq dp ;

B t =∫ B q ,− p ,t dq dp

xj

x j

〈W 〉 j − F ≥ kT ln

 

j F j B

j j 

t0

t

t1

j

j

〈W 〉− F ≥ kT D F ||  B  N

where D Fj || Bj =∑ Fj ln j =1

j

F Bj

Since we don't have full information of the phase densities, we can have only a lower bound. If we have no information at all (N=1), then

D F || B =0  〈W 〉≥F

2nd law!

Overdumped Brownian Particle in a Harmonic Potential



x , t =∫ x , p , x 1, p1, x 2, p 2, ⋯,t dp dx 1 dp1 ⋯

Forward Process

Backward Process

Heat Bath

〈W 〉−F ≥DF x , t ∥B x ,t 

Application: Physics and Information Szilard found a relation between physics and information.

1 bit =k B l o g 2

Leó Szilárd (1898-1964)

Landauer principle

The erasure of one bit of information is necessarily accompanied by a dissipation of at least kBT log 2 heat. Information can be obtained without dissipation of heat.

Q≥k B T log 2

Ralf Landauer (1929-1999) i1 i2

OR

o

Szilard's Engine T

Q  W =k B T ln 2 Contradiction to 2nd Law?

T

T W Q

kT ln2

T

T

Brownian Engine (Backword Process) W =−kT ln 2

b

a

d

c

Brownian Computer (Forward Process)

a

b

W =kT ln 2

Recording

Erasure d

c

Restore-to-One Procedure: d a  b c  d

Coarse Grained Measurement

L

R

PF  x

PB x

1 2

L

1-

 R

 L

[

]

R

P F R  〈W 〉R ≥ln =k B T ln 2k B T ln1− P B R 

For quantum systems

von Neumann Entropy : S=−k Tr  ln   ] 〈 W dis 〉 =kT [ Tr  F ln  F −Tr  F ln  B 

Conclusion  F q , p ,t  〈W 〉−F =k B T ∫  F q , p , t ln dq dp  B q ,− p , t  =k B T D  F ||  B 

An exact expression of dissipation is obtained. Now the second law of thermodynamics is an equality! Dissipation is a direct measure of irreversibility (time's arrow). Even when full information is not available, the formula provides a lower bound of the dissipation The relation between information and physical processes is unambiguously formulated. The Landauer principle is proven.