Information acquisition is an irreversible process. One cannot return to the pristine state of ignorance once apprised any given fact. Heat dissipation also obeys the arrow of time: The heat equation in physics describing its transition is not time symmetric. This paper begins with an observation that this link is not merely philosophical. In static models of Bayesian information acquisition, the value function of beliefs and the quantity of information acquired obeys an inhomogeneous form of the heat equation.

We show that a nonlinear transformation of the value function and beliefs exactly obeys the heat equation. This paper exploits this and a parallel insight and crafts the first global theory of the value of information and its demand. For a binary state world, we derive explicit formulas that provide the bigger picture on the famous nonconcavity of information, and the unique demand curve that it induces: We characterize the "choke-off demand'' level, and also make many novel findings outlined below.

Our key insight is to assume `natural units' corresponding to the sample size of conditionally i.i.d. signals --- focusing on the smooth nearby model of the precision of an observation of a Brownian motion with uncertain drift. In a two state world, this produces the heat equation from physics, and leads to a tractable theory.

At an analytic level, our analysis traces the development of the Black-Scholes option pricing formula. Black and Scholes likewise used the heat equation. Like Harrison and Kreps (1979) attack on the option pricing formula, we find that a martingale approach more rapidly and intuitively arrives at the desired formula.

Among the fundamental properties of Bayesian information demand, we find that: