Krus, D. J., & Webb, J. M. (1994). Maxwell demons: Simulating probability distributions with functions of propositional calculus. Journal of Statistical Computing and Simulation, 51, 71-77.
 

Maxwell’s demons: simulating probability distributions with
functions of propositional calculus

David J. Krus
Arizona State University

James M. Webb
Kent State University

Summary. -A theoretical model of decision-making postulating a threshold/gate mechanism at the decision points of the Galton’s Quincunx, analogous to Maxwell’s demon of statistical mechanics, was used to demonstrate a relationship between probability distributions and theoretical structures of formal logic.

 

A seminal idea in statistical mechanics is that of Maxwell's demon. Named after the Scottish physicist James Clerk Maxwell, Maxwell's demon is a hypothetical homunculus that is considered to admit or block passage of individual molecules between adjacent compartments. If provided with information about the speed of individual molecules, Maxwell's demon would be able to violate the second law of thermodynamics (Tolman, 1955).

The notion of Maxwell's demon can be adapted for use within the theory of statistical probability functions and its associated theories of measurement and scaling. It could be also of interest to theoretical statisticians interested in logical dependencies among variables.

 Let us assume that a group of Maxwell's demons operate within an environment of gates and compartments, provided by Galton's Quincunx (Galton, 1889). Let us further assume that each demon occupies a single decision node in the Quincunx and acts in accordance with the principles of formal logic, as defined by functions of propositional calculus. The experimenter can select functions, determining the demons' behavior, for each experimental run of the Quincunx. In this paper we describe the results of three trial runs, using different logical functions for each trial.

The Quincunx of Maxwell's Demons

Aside from its characteristic honeycomb lattice of decision points connected with bottom compartments characteristic of Galton's Quincunx, the Quincunx of Maxwell's demons also contains an incipient data matrix of all possible responses to a set of binary variables. This data matrix is called plenum and is conceptualized as a truth table of the formal logic. For the discussed trial runs, this plenum contained all possible responses to a set of three binary variables p, q, and r, located on the left side of the following diagrams. On the right side of these diagrams are located matrices of outcomes, containing response patterns congruent with the logical function given to the Maxwell's demons by the experimenter. These logical functions determine the trajectory of a ball moving through the grid of Quincunx's decision points, controlled by a group of Maxwell's demons operating the gate mechanism of the Quincunx in accordance with the principles of Boolean algebra.

Simulating a Binomial Distribution

This simulation corresponds to Galton's original model of the Quincunx. Let us submit to the demons a tautological function

 

The truth table for tautology is

 

and the truth table for the logical function of conjunction is

 

Solving the terms in the parentheses first

 

gives the values of this function as shown in the last column of the above table. The computerized version of Maxwell's demons operates in this case as follows.

 

 

The plenum of responses to a set of binary variables is solved as if it would be a truth table of formal logic. For the discussed case of the binomial distributions, this solution is tautological. All possible response patterns within a given plenum are replicated in the matrix of outcomes. The logic of tautological functions is that anything that is possible within the repertoire of an organism may and will be tried. However, other logical strategies are possible, as discussed in the section to follow.

Simulating a Rectangular Distribution

If we submit to Maxwell's demons a logical function

 

the outcome results in a rectangular distribution. We solve the above function as follows. Recall that the truth table for the logical function of implication is

 

Solving the terms in the parentheses first

 

gives the values of this function as shown in the last column of the above table. Next, submit the solved function  to the Maxwell demons and observe the outcome, shown in the diagram below.

 

During the trial run, the balls moving through the lattice of decision points will form a rectangular distribution in the bottom compartment that corresponds to an idealized version of a perfect Guttman scale, also called the implicatory scale (Krus, 1977).

Simulating Statistical Significance

To simulate a test of statistical significance, the behavior of Maxwell's demons has to be determined by two logical functions:

 

and

 

Solving for the f₁

 

and the f₂

 

The outcome of this simulation is shown in the diagram below.

 

On a trial run, the balls moving through the lattice of the Quincunx decision points will form two binomial distributions. The means of these distributions ( ,  ) are separated by the amount that should be statistically significant. Whether this is the case remains an open question that should be either qualified or answered in the course of the future research.

References

Galton, F. (1889) Natural inheritance. London: Macmillan.

Guttman, L. (1944) A basis for scaling qualitative data. American Sociological Review, 9, 139-150.

Krus, D. J. (1977) Order analysis: An inferential model of dimensional analysis and scaling. Educational and Psychological Measurement, 37, 587-601.

Tolman, R. C. (1955) The principles of statistical mechanics. London: Oxford University Press.