A Thermodynamics of Agency

The five formulas

1. I

   Formula I: Held futures are entropy

   Proved

   $$\pi (G)=\frac{N_{\tau }(G)e^{-U(G)/\alpha }}{Z},\alpha =\frac{\lambda }{\ln b}$$

   $$\ln \frac{\pi (G^{\prime })}{\pi (G)}=\mathrm{\Delta }\ln N_{\tau }-\frac{\mathrm{\Delta }U}{\alpha }$$

   In plain words

   Picture a traveller on a chain of islands whose bridges keep falling and being rebuilt. When the traveller holds a plan, the plan is part of the world, so a world with more open routes can be arranged in more ways. Arrangements that can happen in more ways turn up more often, just as a gas spreads out because spread-out arrangements outnumber the rest. So open futures pull on the world like any other kind of entropy, traded against energy at the world's own temperature.

   G

   the arrangement of the world (which bridges stand)

   N_τ(G)

   the number of τ-step futures the arrangement leaves open

   U(G)

   the energy the arrangement holds, λ per bridge

   α

   the temperature of the store the world exchanges energy with

   New here. The push toward open futures (the causal entropic force) had a temperature set by hand. When futures are held as state, that temperature is the world's own: T = α. Proved for every archipelago and every horizon, for an agent that holds one plan and acts locally and reversibly.

   EvidenceWhen Futures Become EntropyCommitted Futures as MicrostatesManuscript with proofs

2. II

   Formula II: The price of knowing

   Known physics

   $$K(G,y)=\frac{N_{\tau }(G)}{n_y(G)b}$$

   $$W=\alpha I(G),1\text{bit}⟷\alpha \ln 2$$

   In plain words

   A tip about where the future goes is worth exactly what it rules out. Each yes-or-no question it settles is one bit, and a bit has a fixed price in energy: the same price Landauer found for erasing one. In our island world, where each packet of timber doubles the ways its store can be arranged, one bit of foresight trades evenly for one packet.

   K(G, y)

   the odds of trading a correct note for one stored packet

   n_y(G)

   how many of the futures end where the note says

   b

   the store's multiplier per packet

   W, I(G)

   the free energy of a correct record, and the information it carries

   Known physics: Szilard and Landauer's exchange rate. New here is a world in which the controller, its record, and what it controls share one energy account and one temperature, so the price can be checked exactly.

   EvidenceWhat Is a Bit of Foresight Worth?Result record

3. III

   Formula III: The price of keeping the future open

   Known physics

   $$\sigma =J\ln \frac{b_c}{b_h}\ge {\sigma }_{\text{floor}}=-\frac{d}{dt}D(p\Vert {\pi }_c)$$

   $$Y=\frac{H}{J}\le Y_{\text{bound}}=\frac{H\ln \frac{b_c}{b_h}}{{\sigma }_{\text{floor}}}$$

   In plain words

   Left alone, the world forgets: bridges wear away and open futures close. Keeping them open is like heating a house in winter. The heat leaks out, and you keep paying to replace it. There is a minimum bill, set by how fast the world would forget on its own, and nobody can pay less. The agents we measured pay two to three times that minimum.

   J

   the fuel flow: packets per unit time from a hot store, through the world, into a cold one

   σ

   the entropy the agent produces doing it

   σ_floor

   the lowest entropy production any control can approach while holding the same state

   H, Y

   the open future held above the undriven world, and the yield per unit of fuel

   Known physics: the floor is Horowitz, Zhou, and England's (2017). New here is the yardstick Y. It turns agency into a measurable efficiency with a hard bound beneath it. Measured agents pay 2.2 to 3 times the floor.

   EvidenceHow Much Open Future Can a Unit of Energy Buy?Result record

4. IV

   Formula IV: The law of aim

   Approximate

   $$\frac{\partial ⟨N_2⟩}{\partial r_e}=\{\begin{array}{ll}3+(N-2)r& \text{at the agent}\\ \approx 2r& \text{elsewhere}\end{array}$$

   $$E=\frac{H}{H_{\text{warm}}}\approx \frac{N(3+(N-2)r_0)}{2(3+(N-2)(\rho +r_0))}\underset{N\to \mathrm{\infty }}{⟶}\frac{N}{4}$$

   In plain words

   Where energy goes matters more than how much. Fuel spent on the bridges where the traveller stands keeps more future open than the same fuel scattered at random, the way repairing the road you are about to drive beats repairing roads across the country. The bigger the world, the bigger that advantage. A theory with nothing fitted predicted it before it was measured: to a third of a percent when the traveller moves at the world's pace, and less well when it moves faster.

   N

   the number of places in the world

   r, r₀

   bridge density, and its value in the world left to itself

   ρ

   the agent's own mean bridge density, fixed by its fuel flow

   E

   the edge: open future held, over that of random placement at equal fuel flow

   New here, and approximate. A theory with no fitted parameter predicted 32 new edges within 0.36% before they were computed, where the agent moves at the world's own pace. It missed by up to 8.8% when the agent moves sixteen times faster. The N/4 limit holds at weak drive.

   EvidenceDoes Seeing Further Ahead Buy More Open Future?Can a Theory Predict What Energy Buys?The theory

5. V

   Formula V: The information law

   Proved

   $$\sigma \ge {\sigma }_{\text{floor}}(p)-\dot{I}$$

   $${\sigma }_{\text{mem}}\ge \dot{I}+{\dot{I}}_{\text{noise}}$$

   In plain words

   Knowing the world can replace fuel, but only one for one. An agent that knows where the world is slipping can act just there, and each unit of information its actions use saves at most one unit of fuel. Knowing is not free: the memory pays for every bit it learns, including the bits the world's noise wipes out before they are used. Like Maxwell's demon, an agent can even win fuel back, but only because its memory pays the difference. In total, knowledge never beats the minimum; it pays off when it cuts more waste than it costs to gather.

   σ

   the entropy the agent's fuel produces holding the world

   İ

   the information the agent's actions use, per unit time

   σ_mem

   the entropy the agent's memory produces to keep reading the world

   İ_noise

   the information the world's noise erases from the memory, never negative

   Proved here, for every finite world whose own noise does not read the agent's memory, and checked exactly in 500 random worlds. Both lines are exact balances: each becomes an equality once the waste in the actions or in the memory, never negative, is added. The constant is sharp: agents with ever gentler actions reach one nat of fuel per nat of information. The ingredients are known: the second law with information flow (Horowitz and Esposito, 2014) and the floor with a controller (Horowitz, Zhou and England; Horowitz and England, 2017), whose bound on the total is the sum of the two lines. New here: the balance that sets the information an agent uses directly against the floor, and the proof that beating the floor takes a memory that reads the world and actions that read the memory.

   EvidenceCan Knowing the World Replace Fuel?The theoremVerification