Value and information: A profit maximizing strategy for Maxwell's demon

Applications of Lagrange's method of undetermined multipliers in physics and economics suggest some analogies between the two disciplines. By using the work of E. T. Jaynes on information theory, the long standing problem of Maxwell's demon in thermodynamics can be cast as a constrained optimization problem where information and energy are dual concepts. Temperature emerges as the energetic price of information. This approach to the problem of the Maxwell demon highlights the economic features discussed by Bridgman. It also suggests that an analog to temperature (or the price of information) should be present in any optimization problem constrained by costly information. A trivial economic example of this is discussed. The joint view of thermodynamics and economics that motivates this paper goes back to Sadi Carnot's (1824) initial purpose in considering the efficiency of heat engines. Percy Bridgman (1961), Nobel Laureate in 1946 for his work on materials at high temperatures and pressures, had this view in mind in commenting on the second law: It must be admitted, I think, that the laws of thermodynamics have a different feel from most of the other laws of the physics. There is something more palpably verbal about them--they smell more of their human origin. The guiding motif is strange to most of physics: namely, a capitalizing of the universal failure of human beings to construct perpetual motion machines of either the first or the second kind. Why should we expect nature to be interested either positively or negatively in the purposes of human beings, particularly purposes of such an unblushingly economic tinge? Two principle applications of Lagrange's method of undetermined multipliers occur in physics and in economics. In economics, these undetermined multipliers are usually related to prices. Building on the framework of E. T. Jaynes (1957) we can treat thermodynamic systems in terms of constrained optimization and derive temperature as an undetermined multiplier. From this viewpoint and considering an analogy to economics, it becomes apparent that temperature is related to the unit cost of information .in an equilibrium system. This approach provides a new perspective on the paradox of the Maxwell demon that deals directly with the economic features discussed by Bridgman. It also points to new ways of modeling the price of information in economic systems, a problem of considerable importance which has thus far eluded a complete resolution.