[PPT] - Electric network for non-reversible Markov chains joint with Aron PowerPoint Presentation

SLIDE 1

Reversible Irreversible Properties

Electric network for non-reversible Markov chains

joint with ´ Aron Folly

M´ arton Bal´ azs

University of Bristol

School and Workshop on Random Interacting Systems Bath, June 2014.

SLIDE 2

Reversible Irreversible Properties

Reversible chains and resistors Thomson, Dirichlet principles Monotonicity, transience, recurrence Irreversible chains and electric networks The part From network to chain From chain to network Properties Effective resistance What works Nonmonotonicity Dirichlet principle

SLIDE 3

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0.

a x y b

SLIDE 4

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0.

a x y b

Rax Rxy Ryb

a x y b

i 1 V 0 V

Electric resistor network: the voltage u is harmonic (C = 1/R): ux =

y

Cxy

z Cxz

· uy; ua = 1, ub = 0.

SLIDE 5

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0.

a x y b

Rax Rxy Ryb

a x y b

i 1 V 0 V

Electric resistor network: the voltage u is harmonic (C = 1/R): ux =

y

Cxy

z Cxz

· uy; ua = 1, ub = 0.

SLIDE 6

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Thus, Pxy = Cxy

z Cxz

SLIDE 7

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Thus, Pxy = Cxy

z Cxz

= : Cxy Cx . Stationary distribtuion: Cx = µx.

SLIDE 8

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Reversible chains and resistors

Thus, Pxy = Cxy

z Cxz

= : Cxy Cx . Stationary distribtuion: Cx = µx. Notice µxPxy = Cxy = Cyx = µyPyx, so the chain is reversible.

SLIDE 9

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Thomson, Dirichlet principles

Thomson principle:

R Q

1 1

The physical unit current is the unit flow that minimizes the sum

f the ohmic power losses i2R.

SLIDE 10

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Thomson, Dirichlet principles

Thomson principle:

R Q

1 1

The physical unit current is the unit flow that minimizes the sum

f the ohmic power losses i2R.

Dirichlet principle:

R Q

1 V u 0 V

The physical voltage is the function that minimizes the ohmic power losses (∇u)2/R.

SLIDE 11

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Monotonicity, transience, recurrence

The monotonicity property: Between any disjoint sets of vertices, the effective resistance is a non-decreasing function of the individual resistances.

SLIDE 12

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity

Monotonicity, transience, recurrence

The monotonicity property: Between any disjoint sets of vertices, the effective resistance is a non-decreasing function of the individual resistances. can be used to prove transience-recurrence by reducing the graph to something manageable in terms of resistor networks.

SLIDE 13

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy

SLIDE 14

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy

SLIDE 15

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy

SLIDE 16

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy

SLIDE 17

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy

SLIDE 18

Reversible Irreversible Properties The part Chain Network

The part

i R/2 ∗λyx R/2

Voltage amplifier: keeps the current, multiplies the potential.

x y

(ux − i · R 2 ) · λyx − i · R 2 = uy It is clear that λyx = 1 λxy .

SLIDE 19

Reversible Irreversible Properties The part Chain Network

Harmonicity

Rxa/2 Rxa/2 ∗λxa Rxb/2 Rxb/2 ∗λxb

a x b

ux =

y

Dxyγxy Dx · uy with γxy =

λxy = 1

γyx , Dxy = 2γxyCxy (λxy + 1) = Dyx, Dx =

z

Dxzγzx.

SLIDE 20

Reversible Irreversible Properties The part Chain Network

Harmonicity

Rxa/2 Rxa/2 ∗λxa Rxb/2 Rxb/2 ∗λxb

a x b

ux =

y

Dxyγxy Dx · uy with γxy =

λxy = 1

γyx , Dxy = 2γxyCxy (λxy + 1) = Dyx, Dx =

z

Dxzγzx.

γxy =

1 γyx

ux =

y Dxyγxy Dx

· uy Dxy = Dyx

SLIDE 21

Reversible Irreversible Properties The part Chain Network

From network to chain

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0. ux =

y

Dxyγxy Dx · uy, ua = 1, ub = 0.

γxy =

1 γyx

ux =

y Dxyγxy Dx

· uy Dxy = Dyx

SLIDE 22

Reversible Irreversible Properties The part Chain Network

From network to chain

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0. ux =

y

Dxyγxy Dx · uy, ua = 1, ub = 0. Pxy = Dxyγxy Dx .

γxy =

1 γyx

ux =

y Dxyγxy Dx

· uy Dxy = Dyx

SLIDE 23

Reversible Irreversible Properties The part Chain Network

From network to chain

Irreducible Markov chain: on Ω, a = b, x ∈ Ω, hx : = Px{τa < τb} (τ is the hitting time) is harmonic: hx =

y

Pxyhy, ha = 1, hb = 0. ux =

y

Dxyγxy Dx · uy, ua = 1, ub = 0. Pxy = Dxyγxy Dx .

γxy =

1 γyx

Pxy = Dxyγxy

Dx

Dxy = Dyx

SLIDE 24

Reversible Irreversible Properties The part Chain Network

From chain to network

Stationary distribution: Dx = µx. Pxy = Dxyγxy Dx = Dxyγxy µx µxPxy · µyPyx = D2

xy;

µxPxy µyPyx = γ2

xy = λxy.

γxy =

1 γyx

Pxy = Dxyγxy

Dx

Dxy = Dyx

SLIDE 25

Reversible Irreversible Properties The part Chain Network

From chain to network

Stationary distribution: Dx = µx. Pxy = Dxyγxy Dx = Dxyγxy µx µxPxy · µyPyx = D2

xy;

µxPxy µyPyx = γ2

xy = λxy.

Reversed chain: Replace Pxy by ˆ Pxy = Pyx · µy

µx .

Dxy stays, λxy reverses to λyx.

γxy =

1 γyx

Pxy = Dxyγxy

Dx

Dxy = Dyx

SLIDE 26

Reversible Irreversible Properties The part Chain Network

Markovian network

ux =

z

Pxzuz;

z

Pxz = 1 ux ≡ const. is a solution of the network with no external

sources. This is now nontrivial.

SLIDE 27

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Effective resistance

Suppose ua, ub given, the solution is {ux}x∈Ω and {ixy}x∼y∈Ω. Current ia =

x∼a

iax flows in the network at a.

SLIDE 28

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Effective resistance

Suppose ua, ub given, the solution is {ux}x∈Ω and {ixy}x∼y∈Ω. Current ia =

x∼a

iax flows in the network at a. Then ia ua − ub = const. = : Ceff

ab =

1 Reff

ab

.

SLIDE 29

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

What works

... the analogy with P{τa < τb}.

SLIDE 30

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

What works

... the analogy with P{τa < τb}. cap(A, B) = Ceff

AB =

1 Reff

AB

for all sets A, B

SLIDE 31

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

What works

... the analogy with P{τa < τb}. cap(A, B) = Ceff

AB =

1 Reff

AB

for all sets A, B

Theorem

Commute time = Reff · all conductances.

SLIDE 32

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Nonmonotonicity

3/2 3/2 ∗1/5 2/2 2/2 ∗5/13 3/2 3/2 ∗5 2/2 2/2 ∗13/5

R

Reff = 27 14 + 1296 1225R + 2268.

SLIDE 33

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Nonmonotonicity

3/2 3/2 ∗1/5 2/2 2/2 ∗5/13 3/2 3/2 ∗5 2/2 2/2 ∗13/5

R

Reff = 27 14 + 1296 1225R + 2268.

SLIDE 34

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case:

SLIDE 35

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case:

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

SLIDE 36

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

SLIDE 37

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

SLIDE 38

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

Irreversible case (A. Gaudilli` ere, C. Landim / M. Slowik): (i∗

u )xy= Dxy ·

γxyu(x) − γyxu(y)
,

EOhm(i∗

u − Ψ)=

x∼y
i∗

u − Ψxy

2 · Rxy.

SLIDE 39

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

Irreversible case (A. Gaudilli` ere, C. Landim / M. Slowik): Reff

ab =

EOhm(i∗

u − Ψ),

(i∗

u )xy= Dxy ·

γxyu(x) − γyxu(y)
,

EOhm(i∗

u − Ψ)=

x∼y
i∗

u − Ψxy

2 · Rxy.

SLIDE 40

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

Irreversible case (A. Gaudilli` ere, C. Landim / M. Slowik): Reff

ab =

min

Ψ: flow EOhm(i∗ u − Ψ),

(i∗

u )xy= Dxy ·

γxyu(x) − γyxu(y)
,

EOhm(i∗

u − Ψ)=

x∼y
i∗

u − Ψxy

2 · Rxy.

SLIDE 41

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

Irreversible case (A. Gaudilli` ere, C. Landim / M. Slowik): Reff

ab =

min

u:u(a)=1, u(b)=0 min Ψ: flow EOhm(i∗ u − Ψ),

(i∗

u )xy= Dxy ·

γxyu(x) − γyxu(y)
,

EOhm(i∗

u − Ψ)=

x∼y
i∗

u − Ψxy

2 · Rxy.

SLIDE 42

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet

Dirichlet principle

Classical case: Reff

ab =

min

u:u(a)=1, u(b)=0 EOhm(iu),

(iu)xy= Cxy ·

u(x) − u(y)
,

EOhm(iu)=

x∼y

(iu)2

xy · Rxy.

Irreversible case (A. Gaudilli` ere, C. Landim / M. Slowik): Reff

ab =

min

u:u(a)=1, u(b)=0 min Ψ: flow EOhm(i∗ u − Ψ),

(i∗

u )xy= Dxy ·

γxyu(x) − γyxu(y)
,

EOhm(i∗

u − Ψ)=

x∼y
i∗

u − Ψxy