A STATISTICAL NOTE ON SYSTEM TRANSITION INTO EQUILIBRIUM

Junfeng Liu

GCE Solutions, Inc., Bloomington, Illinois 61701, USA. Email: jeff.liu@gcesolutions.com.

ABSTRACT. An enclosed system consisting of multiple types of particles undergoing irregu-

lar motions evolves into and maintain a uniform mixture of particle types (e.g., overall equi-

librium caused by diffusion process). Many historical studies emphasize on quantitatively

describing particle’s motion using mathematics and/or physics rules derived from certain

system parameters. We wish to take a new statistical perspective to illustrate the underly-

ing mechanism which confers and sustains the equilibrium. The observed fact that chaotic

particle motions result in reaching and sustaining system equilibrium (uniform particle type

distribution across the spatial domain) motivates us to summarize a nature’s law of chaotic

mixture considering the resultant allocated particle type distributions across pre-specified

hierarchical traits from observing the stochastic system evolution process. Conditional on a

pair of specified adjacent local spatial regions (region identification as a starting trait) in the

system, such a sort of law helps us to formulate sufficient conditions (from multi-scale view-

points) for reaching overall equilibrium rapidly through involving the relevant hierarchical

temporal transition probabilities as materialized traits and the corresponding particle type

distributions during system evolution.

A number of studies are devoted to quantitative description and prediction of diffusion pro-

cess (R. Brown, Phil. Mag. 4, 161-173, 1828; A. Einstein, Annalen der Physik 322, 549-560,

1905; A. Fick, Poggendorff’s Annalen 94, 59-86, 1855). We take a different perspective by

starting with a simple parameterized simulation. In Figures 1 and 2, a two-dimension space

holds uniformly distributed two types of particles (dark ones within the circle (proportion

p =20%); light ones out of the circle (proportion 80%)). At synchronized steps, particle

motion takes place as a combination of two Gaussian random walks (x and y directions with

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variation 0.05, bouncing off at reaching boundary) and the total number of steps per particle

are 40 and 400, respectively. The sub-system states (snapshot at step 0,2,5) corresponding

to two of the five pre-specified disjoint rectangular regions at simulation termination are

displayed along with the five observed dark particle proportions. Diagram formulates a new

point of view.

1) A set of trait values, Ac = {k|k = 1, . . . , K}, is available at the end of current period

Tc = [Time c − 1,Time c]. For example, K is the number of pre-specified disjoint

regions in the overall space and k stands for each region (Panel “Time 3”, Figure 1).

2) For any kc1, k

c2 ∈ Ac, a set of trait values, Ac−1 = {k|k = 1, . . . , K}, is subsequently

available at the end of period Tc−1 = [Time c− 2,Time c− 1], where k represents the

segment [(k−1)/K, k/K] of the probability domain regarding acquiring kc1 conditional

on {kc1, k

c2} during Tc.

3) For any kc−11 , kc−1

2 ∈ Ac−1, a set of trait values, Ac−2 = {k|k = 1, . . . , K}, is sub-

sequently available at the end of period Tc−2 = [Time c − 3,Time c − 2], where k

represents the segment [(k−1)/K, k/K] of the probability domain regarding acquiring

kc−11 conditional on {kc−1

1 , kc−12 } during Tc−1.

4) So on and so forth, we can exhaustively enumerate vectors of temporal hierarchical

transition probabilities, {(kc1,k

c2) ⊃(kc−1

1 ,kc−12 )⊃· · ·⊃(k0

1,k02)}, after c iterations (from

Time 0 to Time c). ⊃ indicates that the former set contains more particles than the

latter set.

At Time 0, if each set (e.g., k01 or k0

2) in the available pair (e.g., (k01,k

02)) possesses an iden-

tical particle type distribution (Pr(dark)= p), the overall system equilibrium appears at

Time c and sustains afterwards. The number of applicable preceding trait space construc-

tion iterations becomes into c = 1 in Diagram when we reset Time= 0 at the initial system

equilibrium. Each particle is identified in terms of the experienced temporal hierarchical

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transition probabilities (traits, Diagram) which are materialized as a result of the coher-

ent overall system evolution process. Figure 3 maps the Diagram with numbers and types

of particles where the cells in each row identifies the particles with the specific probabil-

ities of reaching the right framed cell (R) out of the two framed cells (L,R) in the next

row. For example, in “Time 2” row, the numbers of particles in the cells are denoted as

(n1,. . .,nK) (K = 5). When the particle type (dark:light) ratio of p : (1 − p) is achieved

in each cells of row “Time 2”, the expected proportions of particles moving into R (row

“Time 3”) conditional on moving into (L ∪ R) (row “Time 3”) are∑K

j=1 nj(j∗/K)∑K

j=1 njfor dark

particles and∑K

j=1 nj(j∗∗/K)∑K

j=1 njfor light particles, respectively. The ratio of the two proportions

is∑K

j=1 nj(j∗/K)∑K

j=1 nj(j∗∗/K)∈ [

∑Kj=1 nj((j

∗∗−1)/K)∑Kj=1 nj(j∗∗/K)

,∑K

j=1 nj((j∗∗+1)/K)∑K

j=1 nj(j∗∗/K)]= [1−

∑Kj=1 nj∑K

j=1 njj∗∗, 1+

∑Kj=1 nj∑K

j=1 njj∗∗], with j∗

and j∗∗ ∈ [j − 1, j], j ≥ 1. If ∀ k ≥ 1, limK→∞

∑kj=1 nj∑Kj=1 nj

= 0, then

∑Kj=1 nj∑K

j=1 njj∗∗<

∑kj=1 nj+

∑Kj=k+1 nj∑K

j=k+1 njj∗∗<(

∑kj=1 nj∑K

j=k+1 nj+ 1) 1

k≤ 1

k.

The relative error between two proportions is bounded by 1/k. Nature’s law of chaotic mix-

ture indicates that, the irregular allocation of subjects restricted to a defined space tends to

transform the original nonhomogeneous subject type distribution into a homogeneous one

with identical empirical distributions across any disjoint pre-specified domains.

Figures 1 and 2 exemplify the pre- and post-equilibrium cases under that regulated evolu-

tion mechanism. We denote the local (e.g., two fused cuboidal blocks, Figure 4) and global

(the container holding the system) region dimensions to be l and L, respectively. If parti-

cles have velocities in the order of v along their sample paths within the system, we may

expect that the required time before observing the overall system equilibrium would be in

the order of L/v from global viewpoints. On the other hand, the required time period before

observing any such local equilibrium may only be in the order of l/v from local viewpoints.

The possible large ratio L/l (depending on our viewpoints) implies that overall uniformity

tends to come into being almost instantly once individual particles undergo free irregular

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motion (e.g., after coffee molecules breaking from powder in the hot water). In Figure 4, the

set of traced irregular sample paths (from moment T0 to Tc) implies chaotic overall transi-

tion conditional on reaching the highlighted subregion at Time c (Diagram, Figure 3). The

post-equilibrium case appeared in Figure 2 takes each particle 400 steps (total variation=2).

Although the local viewpoint seems to need 40 steps (total variation=0.2) (L/l ≈ 10) for par-

ticles to be well mixed, the system does not reach equilibrium after 40 steps. This simulation

in fact lacks a chaotic evolution mechanism. W expect that some other simulation mecha-

nisms (e.g., fractional Brownian motion) also have similar conclusions. Taking a multi-scale

viewpoint, we demonstrate that applying nature’s law of chaotic mixture to the hierarchical

particle population allocation with regard to the conceived trait transition probabilities re-

flects, precedes and gives rise to the appearance and sustaining of overall system equilibrium.

1 R. Brown, Phil. Mag. 4, 161-173 (1828).

2 A. Einstein, Annalen der Physik 322, 549-560 (1905).

3 A. Fick, Poggendorff’s Annalen 94, 59-86 (1855).

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Time 0Diffusion process

Time 1

Time 2 Time 3

Time 0Traced state (region 1)

Time 1

Time 2 Time 3

Time 0Traced state (region 5)

Time 1

Time 2 Time 3

0.0

0.1

0.2

0.3

0.4

Empirical dark particle density (region 1−5)

Figure 1: Diffusion simulation (I) with the number of steps 40. The traced state corresponds

to each of disjoint regions (1 to 5) pictured at Time 3. Overall equilibrium is not reached.

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Time 0Diffusion process

Time 1

Time 2 Time 3

Time 0Traced state (region 1)

Time 1

Time 2 Time 3

Time 0Traced state (region 5)

Time 1

Time 2 Time 3

0.0

0.1

0.2

0.3

0.4

Empirical dark particle density (region 1−5)

Figure 2: Diffusion simulation (II) with the number of steps 400. The traced state cor-

responds to each of disjoint regions (1 to 5) pictured at Time 3. Overall equilibrium is

reached.

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1 · · · K-2 K-1 K

↓ ↓↑ ↑ ↑ ↑ ↑

Timec

1 · · · K-2 K-1 K

↓ ↓↑ ↑ ↑ ↑ ↑

Timec− 1

1 · · · K-2 K-1 K

↓ ↓

· · · · · · · · · · · · · · ·

Timec− 2

1 · · · K-2 K-1 KTime0

↑ ↑ ↑ ↑ ↑

Formulation of temporal hierarchical transition probabilities

Probability [ 0K, 1K] · · · [K−3

K,K−2

K] [K−2

K,K−1

K] [K−1

K,KK]

Diagram: Formulation of temporal hierarchical transition probabilities during system evolu-

tion.

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Tim

e 3

Tim

e 2

Tim

e 1

Tim

e 0

P 0−1/5 1/5−2/5 2/5−3/5 3/5−4/5 4/5−5/5

Temporal hierarchical transition probabilitiesand distributions of particles

Figure 3: Formulation of temporal hierarchical transition probability spaces with empirical

distributions of particle types (c = 3 referring to Diagram).

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Paths traced from current instant

Figure 4: Traced sample paths from current instant (Time c) back to Time 0 with Time c

snapshot at the pre-specified region (two fused cuboidal subregions)

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