# The second law and the advantage of measurement?

### Background

To the best of my knowledge, almost all living systems make use of information obtained from sensory organs. If this result is a reflection of natural selection, then this fact suggests that information provides some advantage to agents.

In this question, let us consider the maximum amount of extracted work as the advantage. Due to Landauer's principle, a measurement needs work to reset a memory. We call the work to reset the memory the memory cost. Hence we define the advantage $$A$$ as follows: $$A=W^{max}-C,$$ where we denote by $$W^{max}$$ the maximum work extracted from the controlled system and by $$C$$ the memory cost.

For simplicity, we assume that the internal energy is constant.

### Related work

The studies to exorcise the Maxwell demon have led to two kinds of second law-like inequalities.

The first one claims that the entropy of the controlled system can decrease when considering information current from the system to the "demon" 1: $$\Delta H(X) + Q \geq \Delta I\tag1.$$ From this equation, we obtain $$W \leq -Q \leq \Delta H(X) - I\tag2,$$ which suggests that $$W^{max}$$ increases by reducing $$I$$. Please note that we assumed to ignore the internal energy change. This inequality serves as a basis to calculate the work in the Szilard engine. You can find inequalities that are the same meaning as (1) and (2) in this paper as Eqs.(5) and (6).

The second one asserts that the upper limit of extractable work is bounded by the entropy production of the controlled system 3: $$W \leq \Delta H(X)\tag3,$$ which serves as a basis to discuss the information reservoir. You can find the original form in Eq. (11) in this paper.

Here, $$X$$ denotes the state of the controlled system in the inequality (1) and the state of fuel such as input tape in the inequality (3), respectively; $$H(X)$$, $$I$$ and $$Q$$ denote the entropy of $$X$$, mutual information between $$X$$ and the measurement device, and the heat transfer from the controlled system. We assume that the process is isothermal with temperature $$T$$ and we set $$k_B T = 1$$.

## Question

Does these inequalities explain why information gives agents the advantage $$A$$ in principle? If yes, why? If no, is there a study that reports a physical law explaining the fundamental advantage of information?

## My opinion

I think the answer is no because of the following reason. The assumption of Landauer's principle would be necessary to defend the second law. Then, the memory cost coincides with $$\Delta I$$. So the upper bound on the extractable work, increased by the effect of information, is cancelled out by the cost of memory. So as long as memory is finite, the first inequality cannot account for the "advantage that information gives the agent". The second inequality is independent of the presence or absence of observation, since it simply states that the upper limit of work increases with the entropy increase of the fuel.

1 Ito, Sosuke, and Takahiro Sagawa. "Information thermodynamics on causal networks." Physical review letters 111.18 (2013): 180603.

3 Boyd, Alexander B., Dibyendu Mandal, and James P. Crutchfield. "Identifying functional thermodynamics in autonomous Maxwellian ratchets." New Journal of Physics 18.2 (2016): 023049.

• The question is quite unclear. What do you mean by "advantage" in the question? How are the mentioned inequalities supposed to provide any "advantage" (some logic behind the statement or a reference would be helpful)? I have heard that systems tend to move in the direction of increase of energy absorption from the environment, but that's quite a different thing. Sep 29 '21 at 17:29
• @Pavlo.B. Thank you very much for your suggestion. I updated the question to add the definition of the "advantage". The inequalities provides the advantage because it determines the upper bound of the work. Sep 29 '21 at 21:50

As you already provided, Landauer's principle provides a lower limit to the Cost a living entity must pay for computation. However, the possible reward is not bounded a priori.

The reason for this is that living agents are not Maxwell demons/Szilard engines trying to eke out a living by seemingly decreasing the entropy of a system or converting information to free energy.

Life abounds on Earth because Thermodynamic free energy abounds on Earth (thanks to the Sun). All life increases the total entropy of the system.

In a simple explanation, imagine a robot connected to an enormous battery, and a switch. The robot can flip the switch with 1 bit of information to receive an arbitrarily large amount of energy. The same principle would apply to an animal given a great supply of extremely calorie-dense food. It only needs to choose to eat, for a great reward.

• I don't think that a physics-based law explains the value of information. As a predator, either you detect prey and reach it before it detects you and escapes, or you do not. As prey, either you detect predators and escape, or you do not. The most relevant theories are probably game theory. Sep 30 '21 at 1:03
• A relevant simplified physical model for thinking about predators and prey is to assume two velocities, $v_1$ and $v_2$, and two endurance times $t_1$ and $t_2$. When the prey begins running at distance $d$, it is caught if $d+v_1 t_1 < v_2 t_2$, assuming the predator is 2. The prey thus maximizes $d$ using information to respond as soon as possible, and the predator minimizes $d$. Sep 30 '21 at 1:38
• Thank you very much for interesting proposal! In my understanding, your proposed model seems require some additional assumptions to quantify the "value of information" such as the "gain" of catching the prey and the "cost" of sensing. If those assumptions are determined only by a physical law, this physical law is definitely what I would like to know in this question. Otherwise, the model becomes a kind of what comes from GT or economics, where the modeler can determine the "law" arbitrarily, in principle. Sep 30 '21 at 4:56
• I don't think the gain can be limited by physical law, referring back to the robot with access to a arbitrarily large battery and arbitrarily large possible gain, which is a counterexample to any theory that would propose some finite limit on the gain. Sep 30 '21 at 4:59
• The point is that there is no intrinsic relationship between information and gain, as you can contrive arbitrary scenarios with arbitrary gain and arbitrary required information to access that gain. So there is not a law, only initial conditions. Sep 30 '21 at 5:46