origin of statistical mechanics: hard spheres (1/3) · pdf fileorigin of statistical...

Origin of Statistical Mechanics: Hard spheres(1/3)

Simulations of quantum many-body systems - part 1

Werner Krauth

Laboratoire de Physique StatistiqueEcole Normale Superieure, Paris, France

11 March 2010.

MechanicsStatistical Mechanics

Direct samplingMarkov-chain samplingMaxwell distribution

Werner Krauth Origin of Statistical Mechanics: Hard spheres (1/3)

References

W. Krauth “Four lectures on computational statisticalphysics Les Houches Lecture 2008 (arXiv:0901.2496)

W. Krauth “Statistical Mechanics: Algorithms andComputations” (Oxford University Press, 2006) (“SMAC”)

www.smac.lps.ens.fr contains information and programs


Paris


Newton’s mechanics

t = 0 t = 1.25

wall collision

t = 2.18 t = 3.12

pair collision

t = 3.25 t = 4.03

t = 4.04 t = 5.16 t = 5.84 t = 8.66 t = 9.33 t = 10.37

Ideal billard...


Chaotic dynamics

Alg. event-disks approximation-free

mean free path of hard disks finite

=⇒ Molecular Dynamics solves all problems!

influence of finite precision ...


Chaotic dynamics - finite precision

32-bit floating point numbers ±a×10b: one bit for the sign, 8bits for the exponent b, 23 bits for the fraction a.

precision ≃ ǫ = 2−23 ≃ 1.2×10−7

double precision: ≃ ǫ = 1×10−16

run different versions to see whether it matters


Chaotic dynamics

(64 bit prec)

... t = 31.76 t = 32.80 t = 33.25 t = 33.32 t = 34.94

(32 bit prec)

... t = 32.34 t = 33.16 t = 33.42 t = 33.87 t = 33.93

hyperbolicity of underlying dynamics...

(stat.)


Observables–projected densities

projected densities yTy (a)

t = 0

a

0 Tt t = T

{y -densityat y = a

}

= ηy (a) =1

T

∑

intersections iwith gray strip

in figure

1

|vy (i)|.

special case of observable

〈O(t)〉T =1

T

∫ T

0dt O(t),


Observables–stroboscopic

Stroboscopic snapshots are easier

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5

t = 6 t = 7 t = 8 t = 9 t = 10 t = 11

〈O(t)〉T ≃1

M

M∑

i=1

O(ti ).

Large-time limit most interesting


Boltzmann’s mechanics–Equiprobability

Equal probability for configurations:

a b

{probability of configuration with[x1, x1 + dx1], . . . , [xN , xN + dxN ]

}

∝ π(x1, . . . , xN)dx1, . . . , dxN

with

π(x1, . . . , xN) =

{

1 if no constraint violated

0 otherwise.


Boltzmann’s mechanics–Equiprobability

Proven for Sinai 2-disk model (1970) (50 p.)

Proven for almost all finite-N systems (disks, spheres)(Simanyi, 2002, 2003)


Ergodicity

Many good reasons to believe in ergodicity, besidesmathematical-physics proofs . . .

In general π(a) = π(E (a)...

Most interesting: Jaynes’ approach (minimum entropy)

Not valid for all finite systems


Equiprobability–hard spheres

Equiprobability for hard disks (direct sampling): generate allconfigurations (legal and illegal), sort out illegal ones..

i = 1 i = 2 i = 3 i = 4 i = 5 i = 6

i = 7 i = 8 i = 9 i = 10 i = 11 i = 12

difficult to do better


Direct sampling for hard spheres (direct-disks)

Direct-sampling for hard disks

procedure direct-disks

1 for k = 1, . . . ,N do

xk ← ran[xmin, xmax]yk ← ran[ymin, ymax]for l = 1, . . . , k − 1 do

{if (dist[xk , xl ] < 2σ) goto 1 (reject—tabula rasa)

output {x1, . . . , xN}——

rejections...


Direct sampling and random deposition

tabula rasa rule....

Monte Carlo

random deposition


Unequal probabilities for random deposition

random deposition gives unequal probabilities

a b c d e f


Acceptance probability of Alg. direct-disks

Only six survivors of run with 106 trials (N = 16, η = 0.3)

i = 84976 506125 664664 705344 906340 909040

Nevertheless, Alg. direct-disks is VIP


Acceptance probability of Alg. direct-disks

Compute acceptance probabilities

procedure direct-disks-any

σ ← 0for k = 1, . . . ,N do

{xk ← ran[0, Lx ]yk ← ran[0, Ly ]

σ ← 12 maxk 6=l [dist[xk , xl ]]

ηmax ← πσ2N/(Lx · Ly ) (limiting density)

output ηmax

——

paccept(η)︸︷︷︸

acceptance rate ofAlg. direct-disks

= 1−

∫η

0dηmax π(ηmax)

︸︷︷︸

integrated histogramof Alg. direct-disks-any

.


Partition function of hard disks

number ofconfigurationswith density 0

: Z (η = 0) =

∫

dx1 . . .

∫

dxN = V N .

number ofconfigurationswith density η

: Z (η) =

∫

. . .

∫

dx1 . . . dxN π(x1, . . . , xN)︸︷︷︸

for disks of finite radius

= Z (0) · paccept(η).


Partition function of hard disks II

Compare acceptance rate with second-virial expansion . . .

N = 16

1

1.e-6

0 0.1 0.2 0.3 0.4

pac

cept(

η)

density η

from π(ηmax)exp[−2(N−1)η]


Markov-chain hard-sphere algorithm

Accepted and rejected moves

a a (+ move) b

a a (+ move) b

Generalizes heliport


Markov-chains (algorithm)


procedure markov-disks

input {x1, . . . , xN} (configuration a)

k ← Nran[1, N]δxk ← (ran[−δ, δ], ran[−δ, δ])if (disk k can move to xk + δxk) xk ← xk + δxk

output {x1, . . . , xN} (configuration b)

——


Markov-chains (algorithm)

i = 1 (rej.) i = 2 i = 3 i = 4 (rej.) i = 5 i = 6

i = 7 i = 8 (rej.) i = 9 (rej.) i = 10 i = 11 i = 12 (rej.)


Markov-chain algorithm


procedure markov-disks

input {x1, . . . , xN} (configuration a)

k ← Nran[1, N]δxk ← (ran[−δ, δ], ran[−δ, δ])if (disk k can move to xk + δxk) xk ← xk + δxk

output {x1, . . . , xN} (configuration b)

——

historic application of Metropolis algorithm (1953)


Observables–4 particles in a box

projected density of four, sixteen particles in box

η

0Ly0

pro

j. d

ens.

ηy

(his

t.)

y−coordinate a

η

0Ly0

pro

j. d

ens.

ηy

(his

t.)

y−coordinate a

non-homogeneous


Alg. markov-disks in a large system

η = 0.48 η = 0.72

Phase transition in two dimension, discovered by Alder &Wainwright (’60s)

Nature of transition still debated


Convergence problems of Alg. markov-disks

i = 0

disk k

... i = 25600000000

same disk

MC better than virial expansions etc.

Convergence problems at high density . . .


Velocities

Monte Carlo: only positions

Molecular dynamics: positions and velocities

. . . apply equiprobability to velocities...

π(v1, . . . , vN) =

{

1 if no constraint violated

0 otherwise.

constraint:

Ekin =1

2m ·

(v21 + · · ·+ v2

N

)(fixed).

Maxwell distribution (1877) ≡ Alg. direct-surface


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