The Physics of Information: From Maxwell to Landauer

Abstract

We summarize recent experimental and theoretical progress achieved in the physics of information. We highlight the intimate connection existing between information and energy from Maxwell’s demon and Szilard’s engine to Landauer’s erasure principle. We focus both on classical and quantum systems and conclude by discussing applications in engineering and biology.

Appendix 1: Stochastic Thermodynamics and Information Energy Cost

When the size of a system is reduced the role of fluctuations (either quantum or thermal) increases. Thus thermodynamic quantities such as internal energy, work, heat, and entropy cannot be characterized only by their mean values but also their fluctuations and probability distributions become relevant and useful to make predictions on a small system. Let us consider a simple example such as the motion of a Brownian particle subjected to a constant external force. Because of thermal fluctuations, the work performed on the particle by this force per unit time, i.e., the injected power, fluctuates and the smaller the force, the larger is the importance of power fluctuations [51,52,53]. The goal of stochastic thermodynamics is just that of studying the statistical properties of the above-mentioned fluctuating thermodynamic quantities in systems driven out of equilibrium by external forces, temperature differences, and chemical reactions. For this reason it has received in the last twenty years an increasing interest for its applications in microscopic devices, biological systems and for its connections with information theory[51,52,53].

Specifically it can be shown that the fluctuations on a time scale τ of the internal energy ΔU_τ, the work W_τ and the heat Q_τ are related by a first principle like equation, i.e.

$$\displaystyle \begin{aligned} \Delta U_\tau = U(t+\tau) - U(t) = \tilde W_\tau-Q_\tau {} \end{aligned} $$

(4)

at any time t.

Furthermore the statistical properties of energy and entropy fluctuations are constrained by fluctuations theorems which impose bounds on their probability distributions (for more details see Ref. [51,52,53]). We summarize in the next section one of them which can be related to information and to Landuer’s bound.

1.1 Estimate the Free Energy Difference from Work Fluctuations

In 1997 [54, 55] Jarzynski derived an equality which relates the free energy difference of a system in contact with a heat reservoir to the pdf of the work performed on the system to drive it from A to B along any path γ in the system parameter space. Specifically, when a system parameter λ is varied from time t = 0 to t = t_s, Jarzynski defines for one realization of the “switching process” from A to B the work performed on the system as

$$\displaystyle \begin{aligned} W = \int_{0}^{t_s} \dot{\lambda}\, \frac{\partial H_{\lambda} [z(t)]} {\partial \lambda} \mathrm{d} t, {} \end{aligned} $$

(5)

where z denotes the phase-space point of the system and H_λ its λ-parametrized Hamiltonian.¹ One can consider an ensemble of realizations of this “switching process” with initial conditions all starting in the same initial equilibrium state. Then W may be computed for each trajectory in the ensemble. The Jarzynski equality states that [54, 55]

$$\displaystyle \begin{aligned} \exp{(-\beta \Delta F)} = \ \langle \exp{(-\beta W)} \rangle, {} \end{aligned} $$

(6)

where 〈⋅〉 denotes the ensemble average, β ⁻¹ = k_B T with k_B the Boltzmann constant and T the temperature. In other words $\langle \exp {[-\beta W_{\mathrm {diss}}]} \rangle = 1$, since we can always write W =  ΔF + W_diss where W_diss is the dissipated work. Thus it is easy to see that there must exist some paths γ such that W_diss ≤ 0. Moreover, the inequality $\langle \exp {x} \rangle \geq \exp {\langle x \rangle }$ allows us to recover the second principle, namely 〈W_diss〉≥ 0, i.e. 〈W〉≥ ΔF.

1.2 Landauer Bound and the Jarzynski Equality

We discuss in this appendix the strong relationship between the Jarzynski equality and the Landauer’s bound. In Box 1.2 we presented the Landauer’s principle as related to the system entropy. Let us consider as a specific example the experiment on the colloidal particle described in Sect. 2.2 [24]. In the memory erasure procedure which forces the system in the state 0, the entropy difference between the final and initial state is $\Delta S = - k_{\text{B}} \ln (2)$. In contrast the internal energy is unchanged by the protocol. Thus it is natural to await $\Delta F = k_{\text{B}}T \ln (2)$. However the ΔF that appears in the Jarzynski equality is the difference between the free energy of the system in the initial state (which is at equilibrium) and the equilibrium state corresponding to the final value of the control parameter: F(λ(τ)) − F(λ(0)). Since the height of the barrier is always finite there is no change in the equilibrium free energy of the system between the beginning and the end of the procedure. Then ΔF = 0, which implies $\left \langle e^{- \beta W_{\text{st}}} \right \rangle = 1$. Thus it seems that there is a problem between the Landauer principle (see Box 1.2) and the Jarzynski equality of Eq. (6).

Nevertheless Vaikuntanathan and Jarzyski [27] have shown that when there is a difference between the actual state of the system (described by the phase-space density ρ_t) and the equilibrium state (described by $\rho ^{\text{eq}}_{t}$), the Jarzynski equality can be modified:

$$\displaystyle \begin{aligned} \left\langle e^{- \beta W_{\text{st}}(t)} \right\rangle _{(x,t)} = \frac{\rho ^{\text{eq}}(x,\lambda (t))}{\rho (x,t)} e^{-\beta \Delta F(t)} {} \end{aligned} $$

(7)

where $\left \langle . \right \rangle _{(x,t)}$ is the mean on all the trajectories that pass through x at time t.

In the experiment presented in Sect. 2.2, the selection of the trajectories where the information is actually erased corresponds to fix x to the chosen final well at the time t = τ. It follows that ρ(0, τ) is the probability of finding the particle in the targeted state 0 at the time τ. Indeed because of the very low energy measured in the protocol thermal fluctuations play a role and the particle can be found in the wrong well at time τ, i.e. the proportion of success P_S of the procedure is equal to ρ(0, τ). In contrast the equilibrium distribution is ρ ^eq(0, λ(τ)) = 1∕2. Then:

$$\displaystyle \begin{aligned} \left\langle e^{- \beta W(\tau)} \right\rangle _{\rightarrow 0} = \frac{1/2}{P_{S}} {} \end{aligned} $$

(8)

Similarly for the trajectories that end the procedure in the wrong well (i.e. state 1) we have:

$$\displaystyle \begin{aligned} \left\langle e^{- \beta W(\tau)} \right\rangle _{\rightarrow 1} = \frac{1/2}{1-P_{S}} {} \end{aligned} $$

(9)

Taking into account the Jensen’s inequality, i.e. $\left \langle e^{-x } \right \rangle \ge e^ {-\left \langle x \right \rangle }$, we find that Eqs. (8) and (9) imply:

$$\displaystyle \begin{aligned} \begin{array}{l} \left\langle W \right\rangle_{\rightarrow 0} \geq k_{\text{B}} T \left[ \ln(2) + \ln(P_S) \right] \\ \left\langle W \right\rangle_{\rightarrow 1} \geq k_{\text{B}} T \left[ \ln(2) + \ln(1-P_S) \right] \end{array} \end{aligned} $$

(10)

Notice that the mean work dissipated to realize the procedure is simply:

$$\displaystyle \begin{aligned} \left\langle W \right\rangle = P_S \times \left\langle W \right\rangle_{\rightarrow 0} + (1-P_S) \times \left\langle W \right\rangle_{\rightarrow 1} \end{aligned} $$

(11)

where $\left \langle . \right \rangle$ is the mean on all trajectories. Then using the previous inequalities it follows:

$$\displaystyle \begin{aligned} \left\langle W \right\rangle \geq k_{\text{B}} T \left[ \ln(2) + P_{S} \ln(P_{S}) + (1-P_{S}) \ln(1-P_{S}) \right] \end{aligned} $$

(12)

which is indeed the generalization of the Landauer’s limit for P_S < 1. In the limit case where P_S → 1, we have:

$$\displaystyle \begin{aligned} \left\langle e^{- \beta W} \right\rangle_{\rightarrow 0} = 1/2 \end{aligned} $$

(13)

Since this result remains approximatively verified for proportions of success close enough to 100%, it explains why in the experiment we find $\Delta F_{\text{eff}} \approx k_{\text{B}} T \ln (2)$.

This result is useful because it strongly binds the generalized Jarzynski equality (a thermodynamic relation) to Landuer’s bound.