Klimontovich's S-Theorem Convergence Criterion

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Description

Function crit.stheorem is applied to a pair of probability vectors or to a pair of vector of counts associated with two states of the same open thermodynamic system in order to check the Klimontovich's S-theorem applicability.

Usage

1
crit.stheorem(distribution0, distribution1)

Arguments

distribution0

vector of counts representing distribution function of state0 of the (open thermodynamic) system

distribution1

vector of counts representing distribution function of state1 of the (open thermodynamic) system

Details

The full text of the theorem and associated algorithm was presented by Klimontovich (1989) and Anischenko (1994).

Let the open thermodynamic system to be characterized by probability density function f(x) and evolving from state0 to state1. Let the state0 be chosen as "chaos". Then the effective Hamiltonian of the system can be written as H_{0}(x) = -lnf_{0}(x) and it must not change at evolution to the state1, namely:

\int {f_{0}(x)H_{0}(x)dx} = \int {f_{1}(x)H_{0}(x)dx}

The latter is "constant Hamiltonian equation" which is not truth for the most of the cases (except the case of identical probability distributions). To make it truth the function f_{0}(x) has to be represented as a subcase of Gibbs distribution for a state of physical chaos, namely:

f^{*}(x) = exp(\frac{F_{0}-H_{0}(x)}{D}), \int {f^{*}(x)dx}=1

where parameter F_{0} is to be defined by procedure of normalization and D is an effective temperature. f_{0}(x) is now to be substituted by the new f^{*}(x) and temperature is to be varied until "constant Hamiltonian equation" becomes truth. The corresponding D value at which system is thermodynamically balanced is now the indicator of the direction of evolution: if D>1 state0 is actually more chaotic (need to be "heated" for energy balance). Otherwise, if D<1, state1 is more chaotic and a case of D=1 takes place where states are identical.

The next step is to check if the system evolution can be followed in opposite direction: from state1 back to state0. State1 is to be chosen as "chaos" and all the steps to be repeated from very beginning. Another effective temperature D_{bck} is to be calculated representing energy balance of the system at its way back, from state1 to state0 . There are 2 scenarios possible:

A.D>1 and D_{bck}<1 or D<1 and D_{bck}>1

B.D<1 and D_{bck}<1 or D>1 and D_{bck}>1

Scenario A (where "way-forward" and "way-back" temperatures have opposite signs) allows for the system evolution to be consistently explained and more chaotic state to be chosen, whereas scenario B makes a direct evolution between states not possible to be followed.

The actual type of the scenario gives an absolute Klimontovich's S-theorem convergence criterion. Let now a null hypothesis be "two sample distributions correspond to the same open thermodynamic system". Null hypothesis is not rejected at scenario A and the general conclusion is "data samples can be considered as outcomes of the same open system and observed differences can be explained by thermodynamic noise". When scenario B is observed, null hypothesis is to be rejected at some level of significance. In order to determine a criterion of rejection the following steps are to be completed:

1. It is assumed the system has probably passed through the medium state2 on its evolution way from state0 to state1. The corresponding probability density function f_{2}(x) can be chosen as

f_{2}(x) = α f_{0}(x)+(1-α)f_{1}(x), {0<=α<=1}, {\int{f_{2}(x)dx}=1}

by varying parameter α until forward-way and back-way effective temperatures have got the opposite signs for all the steps (from state0 to state2 and from state2 to state1 respectively).

2. When α is detected coefficient of consistency is calculated as r^{2} = (2α-1)^{2}. It indicates how far state2 is located from the original states of the system.

3.If r^{2}=1 state2 coincides with one of the original states, i.e. scenario A has actually taken place. If r^{2}=0 state2 is too far from the original states to follow the system evolution and null hypothesis is surely rejected. Finally, if 0<r^{2}<1, the conclusion will depend on significance level chosen.

Value

r2_val

coefficient of consistency, (0<=r2_val<=1)

Author(s)

Vitaly Efremov <vitaly.efremov@dcu.ie>

References

Y.L.Klimontovich. S-theorem. Zeitschrift fur Physik B Condensed Matter. 1987, Volume 66, Issue 1, pp 125-127.

Yu.L.Klimontovich. Problems of open system statistical theory: criteria of relative state ordering at processes of self-organization. 1989. Usp. Fiz. Nauk. v.158(1) (in Russian)

V.S.Anishchenko, T.G.Anishchenko. On the criterion of the relative degree of order for self-oscillating regimes. Illustration of Klimontovich's S-theorem. Proc.SPIE, v.2098, pp.130-136, 1994.

See Also

cxds.stheorem, d1nat, pvalign

Examples

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#completely different bin counts by their (thermodynamic) nature
h0 <- c(0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1)
h1 <- c(1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0)
crit.stheorem(distribution0=h0, distribution1=h1)

#quazi-gaussian probability vectors with equal means & different variances
f0 <- c(0.0,0.1,0.4,0.4,0.1,0.0)
f1 <- c(0.1,0.15,0.25,0.25,0.15,0.1)
crit.stheorem(f0, f1)

#multimodal bin counts
h0 <- c(1,6,1,6,5,1,2)
h1 <- c(1,6,1,1,9,1,2)
crit.stheorem(h0, h1)

#quazi-gaussian bin counts with a shift between means
h0 <- c(2,2,17,6,1,1,1,0)
h1 <- c(2,3,5,7,7,4,1,0)
crit.stheorem(h0, h1)

#example of 2-step analysis with Klimontovich's S-theorem for 2 
# arrays of outcomes {s0,s1}:
s0<-rep(c(1:11,2),256)
s1<-rep(c(2,3,3,4,5,5,5),55)
# step a. Create probability vectors
b<-d1nat(s0,s1,brks=12); b
# step b. Compare samples with Klimontovich's S-theorem
crit.stheorem(b$f0,b$f1)