Introduction

The ever-increasing demand for computational power across all scientific and technological domains has brought a fundamental question to the forefront: what are the ultimate physical limits to computation? While Moore’s Law has historically described the exponential growth in transistor density, the associated energy consumption is becoming a critical bottleneck. Landauer’s principle, formulated in 1961, provided an initial insight by stating that logically irreversible operations, such as erasing a bit of information, must dissipate a minimum amount of energy, $k_B T \ln 2$. However, this principle, in its original form, applies to specific, idealized scenarios.

Stochastic thermodynamics provides a powerful and general framework to analyze the energetics of systems far from equilibrium, making it an ideal tool to investigate the thermodynamic costs of computation in more realistic and complex settings. This field extends classical thermodynamics by considering the fluctuations and probabilistic nature of small systems, which are inherent in modern computational devices. By applying these tools, we can move beyond simple bit erasure and analyze the entropic costs associated with various computational models, from basic logic gates to Turing machines, and understand how architectural choices impact these costs.

This post aims to provide a pedagogical overview of the stochastic thermodynamics of computation. We will delve into:

• Core Concepts of Stochastic Thermodynamics: Introducing Continuous-Time Markov Chains (CTMCs), entropy definitions, and the crucial concepts of entropy flow (EF) and entropy production (EP).
• Information Theory Essentials: Briefly reviewing key quantities like Shannon entropy and Kullback-Leibler (KL) divergence.
• Landauer’s Principle Re-examined: Discussing the standard Landauer cost and the more nuanced concepts of mismatch cost and residual entropy production.
• Logical vs. Thermodynamic Reversibility: Clarifying the often-misunderstood distinction between these two concepts.
• Accounting Conventions: Establishing consistent rules for calculating entropic costs in computational processes.
• Architectural Impacts: Focusing on straight-line circuits to illustrate how the physical arrangement and coupling of computational elements affect overall dissipation.
• Open Questions: Briefly touching upon areas for future research, such as the thermodynamics of more complex computational models like finite automata and Turing machines.

This discussion draws heavily from the comprehensive review by Wolpert arXiv:1905.05669v2, elucidating its key insights and highlighting the physical intuition behind the mathematical formalism.

Stochastic Thermodynamics: The Basics

At the heart of stochastic thermodynamics lies the description of a system’s evolution in terms of probabilities and transitions between its possible states. For many physical systems operating in a thermal environment, particularly those relevant to computation, Continuous-Time Markov Chains (CTMCs) provide the natural mathematical framework.

Continuous-Time Markov Chains (CTMCs)

A CTMC describes a system that can occupy a discrete set of states $X$. The system stochastically jumps between these states, and the probability $p_t(x)$ of finding the system in state $x \in X$ at time $t$ evolves according to a master equation:

$$\frac{dp_t(x)}{dt} = \sum_{x' \neq x} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)]$$

Here, $K_t(x\vert x')$ is the time-dependent transition rate from state $x'$ to state $x$. We can also write this in matrix form as $\frac{d \vec{p}_t}{dt} = \mathbf{K}_t \vec{p}_t$, where $\mathbf{K}_t$ is the rate matrix with off-diagonal elements $K_t(x\vert x')$ and diagonal elements $K_t(x\vert x) = - \sum_{x' \neq x} K_t(x'\vert x)$.

For a time-homogeneous process, $\mathbf{K}_t = \mathbf{K}$. If the process is also ergodic, the system will eventually relax to a unique stationary distribution $\pi$ such that $\mathbf{K} \vec{\pi} = 0$.

Entropy, Entropy Flow, and Entropy Production

The central quantity is the Shannon entropy of the system’s state distribution:

$$S(p_t) = - \sum_x p_t(x) \ln p_t(x)$$

We set $k_B = 1$ for simplicity, so entropy is dimensionless. The rate of change of this entropy can be decomposed into two key components: entropy flow (EF, denoted $\mathcal{Q}$) and entropy production (EP, denoted $\mathcal{E}$):

$$\frac{dS(p_t)}{dt} = \mathcal{Q}_t + \mathcal{E}_t$$

Integrating over the duration of a process from $t_0$ to $t_1$, we get:

$$\Delta S = S(p_{t_1}) - S(p_{t_0}) = \int_{t_0}^{t_1} \mathcal{Q}_t dt + \int_{t_0}^{t_1} \mathcal{E}_t dt = Q_{tot} + E_{tot}$$

Or, using the notation from Wolpert (2019), $Q_{tot} = \mathcal{Q}$ and $E_{tot} = \mathcal{E}$, so we have the fundamental relation:

$$\Delta S = \mathcal{Q} + \mathcal{E}$$

To see how this decomposition arises, we start by taking the time derivative of the Shannon entropy $S(p_t) = - \sum_x p_t(x) \ln p_t(x)$:

$$\frac{dS(p_t)}{dt} = - \sum_x \left[ \frac{dp_t(x)}{dt} \ln p_t(x) + \frac{dp_t(x)}{dt} \right] = - \sum_x \frac{dp_t(x)}{dt} \ln p_t(x) - \sum_x \frac{dp_t(x)}{dt}$$

Since $\sum_x p_t(x) = 1$, its time derivative is zero: $\sum_x \frac{dp_t(x)}{dt} = \frac{d}{dt} \sum_x p_t(x) = \frac{d}{dt}(1) = 0$. Thus, the second term vanishes:

$$\frac{dS(p_t)}{dt} = - \sum_x \frac{dp_t(x)}{dt} \ln p_t(x)$$

Now, substitute the master equation $\frac{dp_t(x)}{dt} = \sum_{x' \neq x} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)]$:

$$\frac{dS(p_t)}{dt} = - \sum_x \sum_{x' \neq x} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln p_t(x)$$

We can symmetrize the sum. Let $J_{xx'}(t) = K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)$ be the net probability flux from $x'$ to $x$. The sum can be rewritten by swapping $x$ and $x'$ in one part of the term:

$$\frac{dS(p_t)}{dt} = - \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln p_t(x) - \frac{1}{2} \sum_{x, x'} [K_t(x'\vert x)p_t(x) - K_t(x\vert x')p_t(x')] \ln p_t(x')$$ $$\frac{dS(p_t)}{dt} = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] (\ln p_t(x') - \ln p_t(x))$$ $$\frac{dS(p_t)}{dt} = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln \frac{p_t(x')}{p_t(x)}$$

We now add and subtract $\ln \frac{K_t(x\vert x')}{K_t(x'\vert x)}$ inside the logarithm:

$$\frac{dS(p_t)}{dt} = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \left( \ln \frac{K_t(x'\vert x)}{K_t(x\vert x')} + \ln \frac{K_t(x\vert x')p_t(x')}{K_t(x'\vert x)p_t(x)} \right)$$

This splits into two terms:

$$\mathcal{Q}_t = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln \frac{K_t(x'\vert x)}{K_t(x\vert x')}$$ $$\mathcal{E}_t = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln \frac{K_t(x\vert x')p_t(x')}{K_t(x'\vert x)p_t(x)}$$

Giving us the desired decomposition: $\frac{dS(p_t)}{dt} = \mathcal{Q}_t + \mathcal{E}_t$.

• Entropy Flow (EF), $\mathcal{Q}_t$: This term quantifies the rate at which entropy is exchanged between the system and its environment (e.g., thermal reservoirs). For a system in contact with a single heat bath at temperature $T$, $\mathcal{Q}_t = \frac{1}{T} \frac{d\langle E \rangle}{dt}_{heat}$, where $\frac{d\langle E \rangle}{dt}_{heat}$ is the rate of heat flow into the system. A common expression for the instantaneous EF rate is:

  $$\mathcal{Q}_t = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln \frac{K_t(x'\vert x)}{K_t(x\vert x')}$$

  This expression is particularly relevant under the condition of local detailed balance, where the ratio of forward and backward rates is related to the energy change: $K_t(x\vert x') / K_t(x'\vert x) = e^{-\Delta E_{x,x'}/T}$. In such cases, EF represents the heat absorbed from the environment, divided by temperature.

• Entropy Production (EP), $\mathcal{E}_t$: This term quantifies the rate of irreversible entropy generation within the system due to its dynamics. It is always non-negative, $\mathcal{E}_t \ge 0$, which is a statement of the second law of thermodynamics. The instantaneous EP rate is given by:

  $$\mathcal{E}_t = \frac{1}{2} \sum_{x, x'} [K_t(x\vert x')p_t(x') - K_t(x'\vert x)p_t(x)] \ln \frac{K_t(x\vert x')p_t(x')}{K_t(x'\vert x)p_t(x)}$$

  To show that $\mathcal{E}_t \ge 0$, let $A_{xx'} = K_t(x\vert x')p_t(x')$ and $B_{xx'} = K_t(x'\vert x)p_t(x)$. Then the expression becomes:

  $$\mathcal{E}_t = \frac{1}{2} \sum_{x, x'} (A_{xx'} - B_{xx'}) \ln \frac{A_{xx'}}{B_{xx'}}$$

  Each term in the sum is of the form $(a-b)\ln(a/b)$. If $a > b > 0$, then $a-b > 0$ and $a/b > 1$, so $\ln(a/b) > 0$. Thus, $(a-b)\ln(a/b) > 0$.

  If $0 < a < b$, then $a-b < 0$ and $0 < a/b < 1$, so $\ln(a/b) < 0$. Thus, $(a-b)\ln(a/b) > 0$.

  If $a=b$, then $a-b=0$ and the term is zero.

  If any $A_{xx'}$ or $B_{xx'}$ is zero, careful consideration of limits is needed, but generally, if $A_{xx'} > 0$ and $B_{xx'} = 0$, the term is effectively infinite unless the forward rate $K_t(x\vert x')$ is zero. However, for physical transitions, rates are typically positive.

  Assuming all $A_{xx'}, B_{xx'} > 0$, each term $(A_{xx'} - B_{xx'}) \ln (A_{xx'}/B_{xx'}) \ge 0$. Therefore, their sum $\mathcal{E}_t \ge 0$. This inequality is related to the non-negativity of the Kullback-Leibler divergence. For instance, we can write it as $\sum_{x,x'} B_{xx'} D(A_{xx'} \vert \vert B_{xx'})$ if we define $D(a\vert \vert b) = (a/b) \ln(a/b) - a/b + 1$ for $a,b > 0$, which is not standard but shows the structure.

  More simply, using the inequality $x \ln x - x \ge -1$ or $x-1 \ge \ln x$, let $x = A/B$. Then $(A-B)\ln(A/B) = B((A/B)-1)\ln(A/B)$.

  Let $y = A/B$. The term is $B(y-1)\ln y$. If $y>0$, $(y-1)\ln y \ge 0$. Since $B = K_t(x'\vert x)p_t(x) \ge 0$, each term in the sum is non-negative.

  EP is zero if and only if the system is in its stationary state $\pi$ and the detailed balance condition $K_t(x\vert x')\pi(x') = K_t(x'\vert x)\pi(x)$ holds for all $x, x'$ (i.e., the system is in thermodynamic equilibrium).

The total entropy production $\mathcal{E} = \int \mathcal{E}_t dt$ over a process represents the total thermodynamic cost or dissipated work. The relation $\Delta S = \mathcal{Q} + \mathcal{E}$ tells us that the change in the system’s own entropy ($\Delta S$) is the sum of the entropy flowed into it from the environment ($\mathcal{Q}$) and the irreversible entropy produced within it ($\mathcal{E}$). Often, in computational contexts, we are interested in minimizing the heat dissipated to the environment (related to $\mathcal{Q}$) for a given logical transformation that fixes $\Delta S$. This requires minimizing $\mathcal{E}$.

Under conditions of detailed balance, $\mathcal{Q}$ is often interpreted as the average heat flow from the system to the reservoir, and $\mathcal{E}$ as the dissipated work (or non-adiabatic entropy production). The framework also allows for the definition of non-equilibrium free energies, such as $F(p_t) = E(p_t) - T S(p_t)$, whose changes can bound the work done on or by the system.

Information Theory Preliminaries

Before delving deeper into the thermodynamics of computation, it’s essential to recall some fundamental concepts from information theory. These quantities will appear frequently when discussing the costs associated with processing and transforming information.

Let $X$ be a random variable with probability distribution $p(x)$.

• Shannon Entropy: As introduced earlier, the Shannon entropy measures the uncertainty or average information content associated with the random variable $X$:

  $$S(X) = S(p) = - \sum_x p(x) \ln p(x)$$

  It quantifies the number of bits (if using log base 2) needed on average to describe the outcome of $X$.

• Joint Entropy: For two random variables $X$ and $Y$ with joint distribution $p(x,y)$, the joint entropy is:

  $$S(X,Y) = - \sum_{x,y} p(x,y) \ln p(x,y)$$

• Conditional Entropy: The entropy of $X$ given $Y$ is:

  $$S(X\vert Y) = S(X,Y) - S(Y) = - \sum_{x,y} p(x,y) \ln p(x\vert Y)$$

  It measures the remaining uncertainty in $X$ when $Y$ is known.

• Kullback-Leibler (KL) Divergence: Also known as relative entropy, the KL divergence measures the dissimilarity between two probability distributions $p(x)$ and $q(x)$ over the same variable $X$:

  $$D(p \vert\vert q) = \sum_x p(x) \ln \frac{p(x)}{q(x)}$$

  It is always non-negative ($D(p\vert\vert q) \ge 0$) and equals zero if and only if $p=q$. Note that it is not symmetric, i.e., $D(p\vert\vert q) \neq D(q\vert\vert p)$ in general.

• Mutual Information: This quantity measures the amount of information that one random variable contains about another. For variables $X$ and $Y$, it is defined as:

  $$I(X;Y) = S(X) - S(X\vert Y) = S(Y) - S(Y\vert x) = S(X) + S(Y) - S(X,Y)$$

  It can also be expressed using KL divergence:

  $$I(X;Y) = D(p(x,y) \vert\vert p(x)p(y))$$

  Mutual information is non-negative and symmetric. It quantifies the reduction in uncertainty about $X$ due to knowing $Y$, or vice-versa, and also measures the degree of statistical dependence between $X$ and $Y$.

• Cross-Entropy: While not as central to the core arguments of Wolpert (2019) that we will focus on, cross-entropy $H(p,q) = - \sum_x p(x) \ln q(x) = S(p) + D(p\vert\vert q)$ is another related measure often encountered in information theory and machine learning.

These quantities, particularly Shannon entropy, KL divergence, and mutual information, are indispensable for quantifying the information processed by computational devices and for defining various entropic costs, as we will see in the following sections.

Landauer’s Principle: Beyond the Basics

Landauer’s original insight was that logically irreversible operations have a thermodynamic cost. Modern stochastic thermodynamics allows us to refine and generalize this principle. Let $\pi$ represent the logical map or function implemented by the physical process, transforming an initial probability distribution $p_{initial}$ over input states to a final distribution $p_{final} = \pi p_{initial}$ over output states.

The Landauer Cost

The second law of thermodynamics states that the total entropy production, $\mathcal{E}$, for any process must be non-negative ($\mathcal{E} \ge 0$). This total entropy production can be seen as the sum of the change in the system’s entropy, $\Delta S = S(p_{final}) - S(p_{initial})$, and the entropy change in its environment (thermal reservoir), $\Delta S_{env}$:

$$\mathcal{E} = \Delta S + \Delta S_{env} \ge 0$$

The entropy change in the environment is related to the heat dissipated by the system, $\mathcal{Q}_{dissipated}$, to the environment at temperature $T$ by $\Delta S_{env} = \mathcal{Q}_{dissipated}/T$. (We are using $k_B=1$, so entropy is dimensionless and temperature $T$ has units of energy. For simplicity, we can further set $T=1$ in many contexts, making $\mathcal{Q}_{dissipated}$ numerically equal to $\Delta S_{env}$).

Substituting this into the second law:

$$\Delta S + \frac{\mathcal{Q}_{dissipated}}{T} \ge 0$$

Rearranging for the dissipated heat:

$$\frac{\mathcal{Q}_{dissipated}}{T} \ge -\Delta S = -(S(p_{final}) - S(p_{initial})) = S(p_{initial}) - S(p_{final})$$

We define the (unconstrained) Landauer cost of the logical map $\pi$ acting on an initial distribution $p_{initial}$ as the reduction in Shannon entropy:

$$\mathcal{L}(p_{initial}, \pi) = S(p_{initial}) - S(p_{final})$$

Thus, Landauer’s principle states that the minimum heat dissipated by the system is:

$$\mathcal{Q}_{dissipated} \ge T \mathcal{L}(p_{initial}, \pi)$$

If we set $k_B T = 1$ as often done in this context for simplicity (making $T=1$ if $k_B=1$), the bound becomes:

$$\mathcal{Q}_{dissipated} \ge \mathcal{L}(p_{initial}, \pi)$$

This means that any logically irreversible operation that reduces the Shannon entropy of the system (i.e., $\mathcal{L} > 0$, such as erasing a bit) must dissipate at least this amount of heat. The minimum dissipation is achieved when the process is thermodynamically reversible (i.e., total system entropy production $\mathcal{E}_{sys}=0$).

To relate this to the notation established earlier in this post, recall that $\Delta S = \mathcal{Q} + \mathcal{E}$, where $\mathcal{Q}$ was the total Entropy Flow (EF) into the system, and $\mathcal{E}$ was the total system Entropy Production (EP, which is non-negative, $\mathcal{E} \ge 0$).

Substituting this into $\Delta S = \mathcal{Q} + \mathcal{E}$ gives:

$$\Delta S = - \frac{\mathcal{Q}_{dissipated}}{T} + \mathcal{E}$$

Rearranging for the dissipated heat per unit temperature:

$$\frac{\mathcal{Q}_{dissipated}}{T} = \mathcal{E} - \Delta S$$

Since the Landauer cost is defined as $\mathcal{L}(p_{initial}, \pi) = S(p_{initial}) - S(p_{final}) = - \Delta S$, we can substitute this to get:

$$\frac{\mathcal{Q}_{dissipated}}{T} = \mathcal{E} + \mathcal{L}(p_{initial}, \pi)$$

This fundamental equation shows that the total heat dissipated (per unit temperature) is the sum of the Landauer cost (the unavoidable cost due to information change) and the system’s entropy production $\mathcal{E}$ (the cost due to the process’s irreversibility). Since $\mathcal{E} \ge 0$, this directly confirms Landauer’s principle: $\mathcal{Q}_{dissipated}/T \ge \mathcal{L}(p_{initial}, \pi)$. The minimum dissipation occurs when $\mathcal{E} = 0$, i.e., the process is thermodynamically reversible. The subsequent discussion of mismatch cost and residual EP details the components that constitute this $\mathcal{E}$.

Mismatch Cost and Intrinsic Dissipation

The total entropy production $\mathcal{E}_{proc}(p)$ for a given physical realization of the map $\pi$ (denoted by proc) acting on an initial distribution $p$ can be decomposed into two main non-negative components. This decomposition, established within stochastic thermodynamics is:

$$\mathcal{E}_{proc}(p) = \mathcal{M}_{proc}(p) + \mathcal{E}_{res;proc}(p)$$

Let’s break down these components:

1. Mismatch Cost ($\mathcal{M}_{proc}(p)$): Any physical process or device (proc) designed to implement a logical map $\pi$ is typically optimized, implicitly or explicitly, for a specific input probability distribution. Let this “designed-for” or “optimal” input distribution be denoted by $q^{proc}$. If the actual input distribution $p$ that the device encounters differs from this $q^{proc}$, an additional thermodynamic cost is incurred due to this discrepancy. This is the mismatch cost, given by the formula:

   $$\mathcal{M}_{proc}(p) = D(p \vert\vert q^{proc}) - D(\pi p \vert\vert \pi q^{proc})$$

   Here, $D(a \vert\vert b)$ represents the Kullback-Leibler (KL) divergence between distributions $a$ and $b$.

   • The term $D(p \vert\vert q^{proc})$ quantifies the initial “surprise” or inefficiency when the process, expecting $q^{proc}$, receives $p$ instead. It measures how distinguishable the actual input $p$ is from the designed-for input $q^{proc}$.
   • The term $D(\pi p \vert\vert \pi q^{proc})$ quantifies the corresponding “surprise” or inefficiency at the output level. It measures how distinguishable the actual output $\pi p$ is from the output $\pi q^{proc}$ that would have resulted if the input had indeed been $q^{proc}$.

   The mismatch cost $\mathcal{M}_{proc}(p)$ is the net portion of the initial input mismatch (as measured by KL divergence) that is converted into entropy production. It represents the thermodynamic penalty for the device operating on an input distribution it wasn’t specifically designed for. By the data processing inequality for KL divergence ($D(p \vert\vert q^{proc}) \ge D(\pi p \vert\vert \pi q^{proc})$), the mismatch cost is always non-negative ($\mathcal{M}_{proc}(p) \ge 0$). The mismatch cost is zero if $p = q^{proc}$, or, in some specific cases, if the map $\pi$ perfectly “transmits” the distinguishability (e.g., if $\pi$ is an identity map or a simple permutation and no new information about the mismatch is lost or gained during processing).

2. Residual Entropy Production ($\mathcal{E}_{res;proc}(p)$): This component represents the entropy production that occurs even if the input distribution perfectly matches the designed-for prior (i.e., if $p = q^{proc}$, which makes $\mathcal{M}_{proc}(p) = 0$). It quantifies the intrinsic irreversibility of the specific physical process chosen to implement the map $\pi$. This cost arises from factors like finite-time operation, friction, non-ideal control protocols, or suboptimal physical construction of the device. Wolpert (2019) shows that $\mathcal{E}_{res;proc}(p)$ can often be expressed as an average, over certain partitions of the state space (called “islands”, denoted $c$), of the minimum entropy production intrinsic to the process when it is fed its designed-for prior $q^{proc}$ within each island: $\mathcal{E}_{res;proc}(p) = \sum_c p(c) \mathcal{E}_{min;proc}(c)$ where $p(c)$ is the probability that the initial state falls into island $c$, and $\mathcal{E}_{min;proc}(c)$ is the minimum EP generated by the process proc if the input is confined to island $c$ and distributed according to $q^{proc}$ within that island. This residual EP is always non-negative ($\mathcal{E}_{res;proc}(p) \ge 0$).

Combining these with the Landauer cost discussed earlier, the total heat dissipated by the system (per unit temperature, assuming $k_B T=1$) is:

$$\mathcal{Q}_{dissipated} = \mathcal{L}(p, \pi) + \mathcal{M}_{proc}(p) + \mathcal{E}_{res;proc}(p)$$

This equation provides a more complete picture of the thermodynamic costs:

• The Landauer cost $\mathcal{L}(p, \pi)$ is the fundamental, unavoidable dissipation tied directly to the change in information content (Shannon entropy) by the logical map itself. It’s a lower bound if the process is perfectly matched to the input and executed reversibly.
• The Mismatch cost $\mathcal{M}_{proc}(p)$ is an additional penalty if the physical device is not optimized for the actual input data it receives.
• The Residual EP $\mathcal{E}_{res;proc}(p)$ is a further penalty due to the inherent irreversibility of the chosen physical implementation, regardless of input matching.

For an ideal scenario, often referred to as an “All-at-Once” (AO) device (Wolpert, Sec 8), it’s assumed that $\mathcal{E}_{res;proc}(p) = 0$. In this case, $\mathcal{Q}_{dissipated} = \mathcal{L}(p, \pi) + \mathcal{M}_{proc}(p)$. If, additionally, the input distribution $p$ matches the device’s prior $q^{proc}$, then $\mathcal{M}_{proc}(p) = 0$, and we recover the simple Landauer bound: $\mathcal{Q}_{dissipated} = \mathcal{L}(p, \pi)$.

The paper (Sec 5.2, Proposition 1) provides equivalent expressions for the EF (which Wolpert calls $\mathcal{Q}$, heat absorbed) that incorporate this decomposition. The key takeaway is that minimizing dissipation requires not only choosing logically efficient operations but also ensuring the physical implementation is well-matched to the input statistics and is executed as close to thermodynamic reversibility as possible.

Logical vs. Thermodynamic Reversibility: A Crucial Distinction

A common point of confusion in discussions about the thermodynamics of computation is the relationship between logical reversibility and thermodynamic reversibility. It is crucial to understand that these are distinct concepts (Wolpert, Sec 6).

• Logical Reversibility: A logical map $\pi: X \rightarrow Y$ is reversible if it is bijective, meaning that for every output $y \in Y$, there is a unique input $x \in X$ such that $\pi(x) = y$. In simpler terms, you can uniquely determine the input if you know the output. If a map is logically reversible, then the input and output spaces must have the same size, and the map merely permutes the states. Consequently, for any distribution $p$ on $X$, the entropy of the output distribution $S(\pi p)$ is equal to the entropy of the input distribution $S(p)$. This means the Landauer cost for any logically reversible map is always zero:

  $$\mathcal{L}(p, \pi_{rev}) = S(p) - S(\pi_{rev} p) = S(p) - S(p) = 0$$

• Thermodynamic Reversibility: A physical process is thermodynamically reversible if it occurs with zero total entropy production ($\mathcal{E}_{proc} = 0$). This means the process is always in equilibrium with its surroundings, and no energy is dissipated due to friction, uncontrolled expansions, or other irreversible phenomena. Achieving perfect thermodynamic reversibility typically requires infinitely slow execution (quasi-static processes).

It is entirely possible for:

1. A logically irreversible map to be implemented by a thermodynamically reversible physical process. For example, the erasure of a bit (e.g., mapping $\{0,1\}$ to $\{0\}$) is logically irreversible. If the initial state is uniformly random ($p(0)=p(1)=1/2$), and the final state is fixed ($S(\pi p)=0$), the Landauer cost is $\mathcal{L} = \ln 2$. This erasure can, in principle, be performed by a physical process that is thermodynamically reversible ($\mathcal{E}_{proc}=0$). In this case, the heat dissipated would be exactly $\mathcal{Q}_{dissipated} = \mathcal{L} = \ln 2$. This is the scenario Landauer originally considered.

2. A logically reversible map to be implemented by a thermodynamically irreversible physical process. Consider a NOT gate, which is logically reversible. Its Landauer cost is $\mathcal{L}=0$. However, any real physical implementation of a NOT gate operating in finite time will likely have some non-zero residual entropy production ($\mathcal{E}_{res;proc} > 0$) due to, for instance, driving currents through resistive elements too quickly. In this case, the heat dissipated will be $\mathcal{Q}_{dissipated} = \mathcal{E}_{res;proc} > 0$, even though the Landauer cost is zero.

This distinction is vital. The Landauer cost $\mathcal{L}(p, \pi)$ sets the minimum possible dissipation for a given logical map $\pi$ acting on input $p$, achievable only if the physical implementation is thermodynamically reversible and perfectly matched to the input (i.e., $\mathcal{E}_{res;proc}=0$ and $\mathcal{M}_{proc}=0$). Any actual physical process will typically dissipate more heat due to these additional sources of irreversibility.

Historically, the lack of non-equilibrium thermodynamic tools sometimes led to confusion between these concepts. Stochastic thermodynamics provides the precise language needed to separate the information-theoretic aspects (captured by $\mathcal{L}$) from the physical implementation details (captured by $\mathcal{M}$ and $\mathcal{E}_{res}$).

Accounting for Entropic Costs in Computation

When we analyze the thermodynamic cost of a computation, the precise definition of “the system” and the conventions used for attributing costs are critical. Computer science definitions of machines (like Turing machines or circuits) don’t inherently specify their physical implementation or the boundaries for thermodynamic accounting. Wolpert (Sec 7) lays out a set of conventions to ensure consistent analysis.

Key explicit conventions often include (Sec 7.1):

• Uniform Hamiltonians: The system’s Hamiltonian is assumed to be uniform (all states have the same energy) at the beginning and end of each computational step or cycle. This simplifies analysis by removing energy differences from Shannon entropy calculations, though non-uniform Hamiltonians can be incorporated.
• Time-Inhomogeneous Processes: The physical processes ($\mathbf{K}_t$) are generally time-inhomogeneous, allowing for external control protocols to drive the computation.
• Temperature: A default temperature (e.g., $k_B T = 1$) is set for the thermal environment.

More subtle are the implicit accounting conventions (Sec 7.2), which dictate which operations contribute to the device’s calculated costs versus costs borne by an external system. Wolpert proposes a “Standard Accounting” convention:

1. Answer Re-initialization: The cost of taking the output of the computational device (say, $X^{OUT}$) and re-initializing some standard variable in the external world to match this output is attributed to the external system, not the device itself. For example, if a circuit computes a bit $b$, the cost to set an external display bit to $b$ is external.

2. Input Re-initialization: The cost of re-initializing the device’s input variable ($X^{IN}$) to some standard state (e.g., all zeros) after the computation, in preparation for a new input, is attributed to the device.

3. New Input Generation: The cost of generating a new input state (which might be correlated with previous inputs or be drawn from a complex distribution) and providing it to the device’s input variable is attributed to the external system.

Under Standard Accounting, the Landauer cost for the computational device performing a single map $\pi$ from its input $X^{IN}$ (with initial distribution $p_0(X^{IN})$) to its output $X^{OUT}$ (with final distribution $p_1(X^{OUT}) = \pi p_0(X^{IN})$) is thus:

$$\mathcal{L}_{device} = S(p_0(X^{IN})) - S(p_1(X^{OUT}))$$

This convention is crucial. For instance, if a device makes an internal copy of its input and that copy persists, it could appear to have a negative Landauer cost for its primary computation if not accounted for carefully. Standard Accounting typically assumes no such persisting copy is available to the device for subsequent operations unless explicitly modeled.

The Importance of These Conventions:

• Comparability: They provide a baseline for comparing the thermodynamic efficiency of different computational architectures or algorithms.
• Avoiding Double Counting/Missing Costs: They clarify where different entropic contributions are tallied, preventing certain costs from being ignored or counted multiple times when analyzing a larger system comprising a device and its environment.
• Defining Device Boundaries: They effectively define the thermodynamic boundary of the “computational device” itself.

One must also be careful with computations that have variable duration or involve iterative processes. The accounting must clearly delineate the costs associated with each cycle or step, including any necessary state re-initializations or information transfers between the device and its environment.

While Standard Accounting provides a default, other conventions can be used, but they must be stated explicitly to ensure clarity and allow for proper interpretation of the calculated entropic costs. The choice of convention can significantly impact the perceived efficiency of a computational process.

The Impact of Architecture: Straight-Line Circuits

Having established the fundamental costs and accounting conventions, we can now explore how the physical architecture of a computational device impacts its thermodynamic efficiency. Straight-line circuits provide a clear example of how constraints on information flow and processing can lead to costs beyond the bare minimum Landauer cost of an equivalent monolithic (All-at-Once, AO) device.

Modeling Circuits (Wolpert, Sec 10.1)

A straight-line circuit is modeled as a directed acyclic graph (DAG) where nodes are logic gates and edges are wires. Key assumptions for the thermodynamic analysis include:

• Standard Accounting Per Gate: The accounting conventions discussed earlier are applied to each gate individually.
• Wires as Identity Gates: Wires are treated as identity gates, potentially with their own (typically zero) Landauer cost and possible mismatch/residual EP if they are noisy or imperfect.
• Physical DAG Construction: The circuit represents a physical system where gates are distinct physical components. Each gate takes inputs only from its parent gates in the DAG and sends outputs only to its child gates.
• Simplifying Assumptions (for initial analysis, Sec 10.2): To isolate architectural effects, it’s often assumed that individual gates operate with zero residual EP ($\mathcal{E}_{res,gate}=0$) and that their designed-for priors ($q_{gate}$) can be chosen arbitrarily (often to match the actual input marginals at that gate to nullify mismatch cost at the gate level).

Landauer Loss and Architectural Constraints (Wolpert, Sec 9 & 10.3)

Consider a circuit $\mathcal{C}$ that implements an overall logical map $\pi_{\mathcal{C}}$ from the circuit’s primary inputs $X^{IN}$ to its primary outputs $X^{OUT}$. An AO device could, in principle, implement the same map $\pi_{\mathcal{C}}$ with a minimal dissipated heat of $\mathcal{L}(p(X^{IN}), \pi_{\mathcal{C}}) + \mathcal{M}_{AO}(p(X^{IN}))$ (assuming $\mathcal{E}_{res,AO}=0$).

However, the circuit architecture imposes constraints: gates operate on local information. If the inputs to different gates are correlated, but the gates operate as if they were independent (a “subsystem process” in Wolpert’s terminology, Sec 9), this can lead to an additional thermodynamic cost. This extra cost, arising from the inability of the constrained architecture to leverage or preserve system-wide correlations in the same way an AO device could, is termed Landauer Loss.

More formally, if a system composed of subsystems A and B evolves such that A and B process information independently ($\pi^{A,B} = \pi^A \pi^B$), the total Landauer cost of this joint operation is $\mathcal{L}_A + \mathcal{L}_B$. The Landauer cost for an AO device performing the same global transformation on $(A,B)$ is $\mathcal{L}_{A,B}$. The Landauer Loss is the difference (Sec 9, Eq. 98):

$$\text{Landauer Loss} = (\mathcal{L}_A + \mathcal{L}_B) - \mathcal{L}_{A,B}$$

This can be shown to be equal to the change in mutual information between A and B due to the constrained process:

$$\text{Landauer Loss} = I_{p_0}(A;B) - I_{\pi p_0}(A;B)$$

To derive this, let $p_0$ be the initial joint distribution over $(A,B)$ and $p_1 = \pi p_0$ be the final joint distribution after the subsystem processes $\pi^A$ and $\pi^B$ act, i.e., $p_1(a,b) = p(\pi^A(a), \pi^B(b))$. Recall the definitions:

• Landauer cost for subsystem A: $\mathcal{L}_A = S(p_0(A)) - S(p_1(A))$
• Landauer cost for subsystem B: $\mathcal{L}_B = S(p_0(B)) - S(p_1(B))$
• Landauer cost for the combined system (A,B) if processed by an AO device implementing the same overall map $\pi$: $\mathcal{L}_{A,B} = S(p_0(A,B)) - S(p_1(A,B))$
• Mutual Information: $I(X;Y) = S(X) + S(Y) - S(X,Y)$

Starting with the definition of Landauer Loss:

$$\text{Landauer Loss} = [S(p_0(A)) - S(p_1(A))] + [S(p_0(B)) - S(p_1(B))] - [S(p_0(A,B)) - S(p_1(A,B))]$$

Rearrange the terms:

$$\text{Landauer Loss} = [S(p_0(A)) + S(p_0(B)) - S(p_0(A,B))] - [S(p_1(A)) + S(p_1(B)) - S(p_1(A,B))]$$

The first bracket is the initial mutual information $I_{p_0}(A;B)$. The second bracket is the final mutual information $I_{p_1}(A;B)$ (or $I_{\pi p_0}(A;B)$).

Thus,

$$\text{Landauer Loss} = I_{p_0}(A;B) - I_{p_1}(A;B) = I_{p_0}(A;B) - I_{\pi p_0}(A;B)$$

where $p_0$ is the initial joint distribution and $\pi p_0$ is the final one. If the independent evolution of A and B destroys correlations between them (i.e., $I_{\pi p_0}(A;B) < I_{p_0}(A;B)$), there is a positive Landauer Loss, meaning the constrained system dissipates more heat than an equivalent AO device, even if each subsystem process $\pi^A$ and $\pi^B$ is thermodynamically reversible.

For a straight-line circuit, the total dissipated heat will be the sum of Landauer costs of individual gates $\sum_g \mathcal{L}_g$, plus their mismatch costs $\sum_g \mathcal{M}_g$ and residual EPs $\sum_g \mathcal{E}_{res,g}$. The Circuit Landauer Loss is the amount by which $\sum_g \mathcal{L}_g$ (the sum of local Landauer costs) exceeds the global Landauer cost $\mathcal{L}(p(X^{IN}), \pi_{\mathcal{C}})$:

$$\Delta \mathcal{L}_{\mathcal{C}}(p) = \left( \sum_g \mathcal{L}_g(p_{pa(g)}) \right) - \mathcal{L}(p(X^{IN}), \pi_{\mathcal{C}})$$

where $p_{pa(g)}$ is the distribution of inputs to gate $g$. This loss is non-negative and represents the penalty for not being able to process information in an integrated, AO fashion.

Wolpert (Sec 10.3, Proposition 4) provides an expression for the Circuit Landauer Loss for a circuit implementing a Boolean formula, relating it to the difference between the “multi-information” (a measure of total correlation) at the circuit inputs and the sum of multi-informations at the inputs of each gate. Multi-information $\mathcal{I}(p(Z_1, ..., Z_n)) = \sum_i S(p(Z_i)) - S(p(Z_1, ..., Z_n))$ measures the total amount of correlation in a set of variables. The proposition states:

$$\Delta \mathcal{L}_{\mathcal{C}}(p) = \mathcal{I}(p(X^{IN})) - \sum_g \mathcal{I}(p_{pa(g)})$$

This highlights that if the circuit architecture breaks down global input correlations into less correlated local inputs for each gate, a Landauer Loss is incurred.

Similarly, Circuit Mismatch Loss can occur if the gate priors are chosen suboptimally relative to an optimal AO prior (Sec 10.3, Proposition 5). Interestingly, this mismatch loss can sometimes be negative, meaning a constrained circuit can sometimes have a lower mismatch cost than a particular AO device, if the AO device is mismatched to the true input distribution in a particularly bad way that the circuit architecture mitigates.

The key insight is that the very structure of a computational device—how it is partitioned into interacting subsystems—has direct thermodynamic consequences, even if all components are individually optimized. These architectural costs are purely information-theoretic and arise from how correlations are handled (or ignored) by the constrained dynamics.

Further Considerations and Open Questions

The framework of stochastic thermodynamics offers powerful tools to analyze the energetic costs of computation, but many fascinating questions and complex systems remain topics of ongoing research. While we have focused on foundational concepts and straight-line circuits, the principles discussed extend to a broader range of computational paradigms, each with its own unique thermodynamic considerations.

• Finite Automata (FA) and Transducers: Section 12 of Wolpert (2019) discusses the entropy dynamics of FAs, which, unlike circuits, can handle inputs of arbitrary length and involve cycles and memory. Analyzing FAs often involves considering their steady-state behavior or the thermodynamics of processing sequences of symbols. Early work by Chu et al. (2018) (arXiv:1806.04875) explores these aspects, for example, by using time-homogeneous rate matrices and energy landscapes to drive FA transitions.

• Turing Machines (TMs) and Kolmogorov Complexity: While TMs represent the canonical model of universal computation, a full thermodynamic analysis is exceptionally complex. The connection between TMs and Kolmogorov complexity (the length of the shortest program to produce a given output, Sec 4.5 of Wolpert) hints at deep links between minimal program length and minimal thermodynamic cost, but formalizing this is a major challenge. How do the costs of running a Universal Turing Machine, or the costs associated with halting (or not halting), manifest thermodynamically?

• Information Ratchets: These devices, discussed in Section 13 of Wolpert (2019), operate by rectifying thermal fluctuations to perform work or process information, often in response to a stream of inputs that may be non-Markovian or have long-range correlations. Their continuous operation and interaction with input/output streams pose unique analytical challenges.

• Logically Reversible Computing Revisited: While the Landauer cost of a logically reversible operation is zero, Section 11 of Wolpert (2019) delves into the subtleties that arise when considering the full cycle of computation, including initializing ancillary bits and, crucially, re-initializing the copied input and control bits back to a standard state for reuse. When these unavoidable re-initialization costs are factored in, the purported thermodynamic advantages of reversible computing over optimally designed irreversible AO devices are often diminished or eliminated, especially concerning the fundamental Landauer cost of the overall input-output transformation.

• Minimizing Total Dissipation: A central goal is to minimize $\mathcal{Q}_{dissipated} = \mathcal{L}(p, \pi) + \mathcal{M}_{proc}(p) + \mathcal{E}_{res;proc}(p) + \Delta\mathcal{L}_{architecture}$. This requires co-designing the logical algorithm (to minimize $\mathcal{L}$), the physical device (to match $p$ and minimize $\mathcal{M}$), the operational protocol (to minimize $\mathcal{E}_{res}$), and the system architecture (to minimize $\Delta\mathcal{L}_{architecture}$). This holistic optimization is a formidable task.

• Beyond Classical Digital Computation: The Wolpert review deliberately focuses on classical, digital computational machines from computer science theory. However, the principles of stochastic thermodynamics are also being applied to:

  • Quantum Computation: Analyzing the thermodynamics of quantum gates, quantum error correction, and quantum annealing presents unique challenges and opportunities, involving quantum entropy definitions and open quantum system dynamics.
  • Analog and Neuromorphic Computing: These paradigms often rely on continuous state variables and different noise models, requiring extensions of the current framework.
  • Biochemical Computation: The information processing occurring within living cells (e.g., DNA replication, protein synthesis, signaling pathways) is subject to thermodynamic costs. Understanding these from a stochastic thermodynamics perspective is a vibrant field, linking information theory, non-equilibrium physics, and biology.

This field is rich with theoretical challenges and potential practical implications as the demand for energy-efficient computation continues to grow. The detailed understanding of how information, thermodynamics, and physical implementation intertwine is key to pushing the boundaries of what is computationally achievable within the constraints imposed by the laws of physics.

Conclusion

The stochastic thermodynamics of computation provides a rigorous and insightful framework for understanding the fundamental energetic costs associated with information processing. By moving beyond idealized scenarios, it allows us to dissect the various contributions to heat dissipation in realistic computational systems. We have seen that the total thermodynamic cost is not solely dictated by the information-theoretic Landauer cost associated with a logical map. Instead, it encompasses penalties arising from mismatches between the device’s design and actual input statistics (Mismatch Cost), inherent irreversibilities in the physical implementation (Residual Entropy Production), and constraints imposed by the system’s architecture that prevent globally optimal processing of information (Landauer Loss).

Key takeaways include:

• The clear distinction between logical and thermodynamic reversibility, and the understanding that zero Landauer cost (for logically reversible operations) does not imply zero heat dissipation.
• The critical role of accounting conventions in defining the boundaries of the computational device and attributing entropic costs consistently.
• The insight that architectural choices are thermodynamically significant, influencing how well a system can leverage or preserve correlations, thereby affecting overall efficiency.

Minimizing the energy footprint of computation requires a holistic approach, optimizing not just algorithms at a logical level, but also the physical hardware, the control protocols, and the overall system architecture in light of the statistical nature of the data being processed. The tools and concepts from non-equilibrium statistical physics, particularly stochastic thermodynamics, are indispensable for this endeavor.

As computational demands continue to surge, the insights gleaned from the stochastic thermodynamics of computation will be increasingly vital in guiding the design of future information technologies and in deepening our understanding of the intricate relationship between energy, information, and the physical world.