Thermodynamics of Information Part Icsc.ucdavis.edu/~chaos/courses/poci/Lectures/Lecture37a...Energy...

Thermodynamics of InformationPart IAlec Boyd

Thermal Reservoir

Information Engine

Q MassW

Information Reservoir

What does information do?

H(X|Y ) I(X : Y ) H(Y |X)

H(X) H(Y ) How do we use the stuffin these bubbles?

Thermodynamics treats physical systems many elements.

Relevant quantities:1) Energy

Thermodynamics

Relevant quantities:1) Energy2) Work

Thermodynamics

Relevant quantities:1) Energy2) Work3) Heat

Thermodynamics

Relevant quantities:1) Energy2) Work3) Heat4) Temperature

Thermodynamics

Relevant quantities:1) Energy2) Work3) Heat4) Temperature5) Entropy

Thermodynamics

Physical system states:

Thermodynamic Entropy

X = {x}

Probability of physical state at time :

X = {x}

Pr(Xt = x)

Thermodynamic entropy (Gibbs) of physical system:

X = {x}

Pr(Xt = x)

] = �k

x2XPr(X

= x) lnPr(Xt

Shannon entropy of physical system:

X = {x}

Pr(Xt = x)

] = �k

x2XPr(X

= x) lnPr(Xt

] = �X

x2XPr(X

= x) log2 Pr(Xt

Shannon entropy of physical system:

X = {x}

Pr(Xt = x)

] = �k

x2XPr(X

= x) lnPr(Xt

] = �X

x2XPr(X

= x) log2 Pr(Xt

S[Xt] = kB ln 2H[Xt]

Entropy Comparison

Thermodynamic entropy as a measure of disorder.

Disordered (Hot Gas) Ordered (Cold Solid)

Entropy Comparison

Shannon entropy as a measure of uncertainty or unpredictability.

Entropy Comparison

Shannon entropy as a measure of uncertainty or unpredictability.

String of measurements of unpredictable variable

String of measurements of predictable variable

...110111011001011110000000001... ...1010101010101010101010101...

Physical states described by positions and their conjugate momenta :

Classical Physics

X = {x} = {(q1, q2, ..., qN , p1, p2, ..., pN )}

Can be thought of as a collection of particles:

Classical Physics

X = {x} = {(q1, q2, ..., qN , p1, p2, ..., pN )}

(qi, qi+1, qi+2)

(pi, pi+1, pi+2)

Can be thought of as a collection of particles:

How does this change with time?

Classical Physics

X = {x} = {(q1, q2, ..., qN , p1, p2, ..., pN )}

(qi, qi+1, qi+2)

(pi, pi+1, pi+2)

First Law of Thermodynamics

Energy of each state specified by Hamiltonian function of state

E(x) = H(~q, ~p)

Position changes as velocity and momentum changes as the force:

E(x) = H(~q, ~p)

= @piH(~q, ~p)dpidt

= �@qiH(~q, ~p)

Position changes as velocity and momentum changes as the force:

Equations of motion conserve energy

E(x) = H(~q, ~p)

= @piH(~q, ~p)dpidt

= �@qiH(~q, ~p)

�t!t+⌧E = E(x(t+ ⌧))� E(x(t)) = 0

dE(x(t))

Entropy increases in isolated systems:

Second Law of Thermodynamics

dS[Xt]

dt� 0

dH[Xt]

dt� 0

dS[Xt]

dt� 0

dH[Xt]

dt� 0

Ordered (Hot Solid)

dS[Xt]

dt� 0

dH[Xt]

dt� 0

Disordered (Hot Gas) Ordered (Hot Solid)

Example another and another and another

dS[Xt]

dt� 0

dH[Xt]

dt� 0

Disordered (Hot Gas) Ordered (Hot Solid)

Energy and Entropy

According to the first law and second law of thermodynamics together, the microstates evolve towards the maximum entropy distribution over a fixed energy , called the microcanonical ensemble:

Energy and Entropy

where is the number of states with energy .

Pr(Xeq = x) = �

E(x),E/⌦(E)

⌦(E)

Energy and Entropy

where is the number of states with energy .

This leads to an equilibrium estimate of entropy:

Pr(Xeq = x) = �

E(x),E/⌦(E)

⌦(E)

S[Xeq|E(x) = E

0] = kB ln⌦(E0)

Energy/Entropy vs. Temperature

What is temperature?-Average energy?

Temperature of a system in equilibrium:

=@S[Xeq|E(x) = E

System Q

=@S[Xeq|E(x) = E

�Ssys =Q

In most systems, every degree of freedom stores energy proportional to the temperature:

System Q

=@S[Xeq|E(x) = E

hEqii ⇡ hEpii ⇡kBT

�Ssys =Q

Heat Flow

Entropy explains why heat diffuses from hot to cold:

Q THTC

TH > TC

Heat Flow

Total entropy increases by heat diffusion:

Q THTC

�Stotal

= Q/TC �Q/TH > 0

TH > TC

Heat Flow

The reverse is impossible:

Q THTC

�Stotal

= Q/TC �Q/TH > 0

TH > TC

Heat Flow

The reverse is impossible:

Q THTC

�Stotal

= Q/TC �Q/TH > 0

�Stotal

= Q/TH �Q/TC < 0

TH > TC

Heat Engine

Some energy can be redirected to produce work

Heat Engine

Qc Qc + W = Qh

Heat Engine

Entropy increase constrains efficiency

Qc Qc + W = Qh

4Stotal

Heat Engine

Qc Qc + W = Qh

4Stotal

� 0Qc

Qh� Tc

Heat Engine

Qc Qc + W = Qh

e =output

Qh= 1� Qc

Heat Engine

Qc Qc + W = Qh

e 1� Tc

e =output

Qh= 1� Qc

Thermodynamics of Control

How do we extract work from a thermodynamic system?

!Thermal

Reservoir MassWQ

Feedback/Control

Through control of the energy landscape

dhEi =X

= x)dE(x) +X

E(x)dPr(Xt

!Thermal

Reservoir MassWQ

Feedback/Control

dhEi =X

= x)dE(x) +X

E(x)dPr(Xt

= hdW i+ hdQi

!Thermal

Reservoir MassWQ

Feedback/Control

We consider control of a system which interacts with an ideal thermal reservoir at temperature to produce work

dhEi =X

= x)dE(x) +X

E(x)dPr(Xt

= hdW i+ hdQi

!Thermal

Reservoir MassWQ

Feedback/Control

Entropy of Control

The total entropy production of the joint reservoir system

!Thermal

Reservoir Mass

Feedback/Control

�Stotal

= �S[R] +�S[X]��I[X;R]

Entropy of Control

The fact that the reservoir is ideal means it has no memory of the ratchet, and so is uncorrelated with it:

!Thermal

Reservoir Mass

Feedback/Control

�Stotal

= �S[R] +�S[X]��I[X;R]

I[X;R] = 0

Entropy of Control

Heat production in control is bounded by change:

!Thermal

Reservoir Mass

Feedback/Control

�Stotal

= �S[R] +�S[X]��I[X;R]

I[X;R] = 0

+�S[X] = �Stotal

Entropy of Control

Heat production in control is bounded by change:

!Thermal

Reservoir Mass

Feedback/Control

�Stotal

= �S[R] +�S[X]��I[X;R]

I[X;R] = 0

+�S[X] = �Stotal

hQi � �T�S[X]

Entropy of Control

This can be re-expressed with work

!Thermal

Reservoir Mass

Feedback/Control

T�Stotal

= hW i ��hE[X]i+ T�S[X]

Entropy of Control

This can be interpreted as dissipated work

!Thermal

Reservoir Mass

Feedback/Control

hWdissi = hW i ��FNEQ

T�Stotal

Entropy of Control

This can be interpreted as dissipated work

This includes a new quantity: the non-equilibrium free energy, which specifies the amount of work that could have been harvested:

!Thermal

Reservoir Mass

Feedback/Control

hWdissi = hW i ��FNEQ

FNEQ = hE[X]i � TS[X]

T�Stotal

Stepping Back

What does entropy do?

Stepping Back

1) Bounds function of heat engine

e 1� Tc

e =output

Qh= 1� Qc

Stepping Back

1) Bounds function of heat engine with Carnot efficiency

e 1� Tc

e =output

Qh= 1� Qc

Stepping Back

2) Bounds work and heat production in controlled thermodynamic systems

e 1� Tc

e =output

Qh= 1� Qc

Stepping Back

hQi � �kBT ln 2�H[X]

e 1� Tc

e =output

Qh= 1� Qc

Stepping Back

hW i � h�E[X]i � kBT ln 2�H[X]

e 1� Tc

e =output

Qh= 1� Qc

Control Example

Single particle in a box

Control Example

Single particle in a box, insert barrier

Control Example

Single particle in a box, insert barrier, slide barrier away from particle

Control Example

Single particle in a box, insert barrier, slide barrier away from particle, while in contact with heat reservoir at temperature T

Thermal Reservoir Q

Control Example

Single particle in a box, insert barrier, slide barrier away from particle, while in contact with heat reservoir at temperature , and extract work with the piston.T

Thermal Reservoir Q MassW

Control Example

Single particle in a box, insert barrier, slide barrier away from particle, while in contact with heat reservoir at temperature , and extract work with the piston.

How much much work and heat flow?

Thermal Reservoir Q MassW

hW i � h�E[X]i � kBT ln 2�H[X]

Control Example

Initial vs. final energies and entropies:

Control Example

Energy of each position is the same, so average energy of positions is the same: hE~qiinitial = hE~qifinal

Control Example

Energy of each position is the same, so average energy of positions is the same:If in equilibrium, average kinetic energy of momentum variable is the same:

hE~qiinitial = hE~qifinal

hE~piinitial = hE~pifinal =kBT

2= hKinetic Energyi

Control Example

Energy of each position is the same, so average energy of positions is the same:If in equilibrium, average kinetic energy of momentum variable is the same:

Average energy doesn’t change:

hE~qiinitial = hE~qifinal

hE~piinitial = hE~pifinal =kBT

2= hKinetic Energyih�E[X]i = 0

Control Example

Vinitial

Vfinal

Control Example

Probability of each position is uniform and inversely proportional to the available volume:

Pr( ~Qfinal = ~q) =c

VfinalPr( ~Qinitial = ~q) =

Vinitial

Vfinal

Control Example

At equilibrium, the probability of momentum is the same initially and finally

Vinitial

Vfinal

Pr(~Pinitial = ~p) = Pr(~Pfinal = ~p) = Pr(~P eq = ~p)

Control Example

At equilibrium, the probability of momentum is the same initially and finally, and independent of position distribution.

Vinitial

Vfinal

Pr(~Pinitial = ~p) = Pr(~Pfinal = ~p)

Pr(Xinitial = (~q, ~p)) =cPr(~Pinitial = ~p)

VinitialPr(Xfinal = (~q, ~p)) =

cPr(~Pfinal = ~p)

Vfinal

= Pr(~P eq = ~p)

Control Example

Thus, the change in Shannon entropy is

Vinitial

Vfinal

�H[X] = H[Xfinal]�H[Xinitial]

Control Example

Vinitial

Vfinal

~P eq]�

x2Vfinal

Vfinallog2

Vfinal�H[

~P eq] +

x2Vinitial

Vinitiallog2

Vinitial

Control Example

Vinitial

Vfinal

~P eq]�

x2Vfinal

Vfinallog2

Vfinal�H[

~P eq] +

x2Vinitial

Vinitiallog2

Vinitial

= log2Vfinal

Vinitial

Control Example

And the work and heat are bounded by the log ratio of the volumes (achievable if done very slowly):

Vinitial

Vfinal

~P eq]�

x2Vfinal

Vfinallog2

Vfinal�H[

~P eq] +

x2Vinitial

Vinitiallog2

Vinitial

hW i � �kBT ln 2 log2Vfinal

VinitialhQi � �kBT ln 2 log2

Vfinal

Vinitial

= log2Vfinal

Vinitial

Szilard’s Information Engine

Take a particle in a box:

Insert a barrier at halfway:

Measure which side the particle lands on:

left right

Use measurement to slowly slide barrier away from particle, extracting work:

left right

Use measurement to slowly slide barrier away from particle, extracting work:

Store work, repeat:

left right

hW i = �kBT ln 2

Maxwell’s Demon

http://www.eoht.info/page/Maxwell's+demon

Maxwell’s Demon

!Thermal

Reservoir MassWQ

Maxwell’s Demon Feedback/Control

Maxwell’s Demon

!Thermal

Reservoir MassWQ

hW i = hQi = �kBT ln 2

Maxwell’s Demon

!Thermal

Reservoir MassWQ

�H[X] = 0

Maxwell’s Demon

!Thermal

Reservoir MassWQ

�H[X] = 0

�Stotal

+ kB ln 2�H[X]

Maxwell’s Demon

!Thermal

Reservoir MassWQ

�H[X] = 0

�Stotal

+ kB ln 2�H[X]

= �kB ln 2 < 0

Maxwell’s Demon

!Thermal

Reservoir MassWQ

�H[X] = 0

�Stotal

+ kB ln 2�H[X]

= �kB ln 2 < 0 ?

Maxwell’s Demon

!Thermal

Reservoir MassWQ

�Stotal

Maxwellian Demons

Measure

Control repeat

Heat Bath Temp=T

System Under Study (SuS)

Closed System

Imeasure

Icontrol

Szilard Engine Cycle

measure

control

measure

control

measure

Demon is able to extract and store work every cycle.

kBT ln2

control

measure

control

measure

Demon is able to extract and store work every cycle.

kBT ln2

control

A Demon is able to extract and store work every cycle.

Violates the second law?

measure

control

measure

kBT ln2

control

Energetic Costs

Qerase � kBT ln 2Landauer’s principle:R. Landauer, “Irreversibility and heat generation in the computing process”, (1961).

0 1 0 1

Energy

ErasureErasure

4E > kBT

Energy Energy

Erasure:(resetting memory)

Measurement:(writing to memory)

L. Brillouin “Maxwell’s Demon Cannot Operate”, (1951).

� ⇡ hc

Information Bearing Degrees of Freedom

!Thermal

Reservoir MassWQ

�Stotal

!Thermal

Reservoir MassWQ

Erasure

?Thermal

Reservoir MassWQ

�Stotal

!Thermal

Reservoir MassWQ

Erasure

?Thermal

Reservoir MassWQ

Landauer’s Principle: Werase

� kBT ln 2 ! �Stotal

�Stotal

Benefit of Information Processing

Information processing allows systems to leverage structure of environment

http://spectrum.ieee.org/aerospace/robotic-exploration/the-nearly-effortless-flight-of-the-albatross

Cost of Information Processing

http://pooh.wikia.com/wiki/Think,_Think,_Thinkhttp://spectrum.ieee.org/computing/hardware/new-tech-keeps-data-centers-cool-in-warm-climates

Processing information costs work and dissipates heat.

Szilard Example

The Szilard system can be reformulated as a 2D box of gas.

SuS Memory Macrostates

Demon Memory Macrostates

RightLeft

C. Bennet, “Thermodynamics of Computation-A Review”, (1981)

http://arxiv.org/abs/1506.04327

Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems

Szilard Engine Dissipation

The average dissipated heat is exactly calculable as the work done to slide the barriers:

Q = W = �Z

= �T�S[X]

Q = W = �Z

= �T�S[X]

�Stotal

hQerasei hQmeasurei

hQcontrol

hQdissi[k

A. Boyd, J. Crutchfield, “Maxwell Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems”, PRL, (2016)

Information is a Thermodynamic Fuel

Thermal Reservoir MassWQ

Instead of erasing, write to a hard drive

Thermodynamics of Information Part Icsc.ucdavis.edu/~chaos/courses/poci/Lectures/Lecture37a...Energy...

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