Is Stochastic Thermodynamics the Key to Understanding the Energy Costs of Computation?

Wolpert, D. H. et al. Is stochastic thermodynamics the key to understanding the energy costs of computation? Proceedings of the National Academy of Sciences 121, e2321112121 (2024).

When discussing the energetic cost of computation, one of the most common starting points is Landauer’s principle: erasing one bit of information requires at least a thermodynamic cost of $k_BT \ln 2$ . This statement is attractive because it compresses the relation between information and energy into a simple formula. If computation involves information erasure, heat must be dissipated. In this sense, computation seems to have a physical lower bound.

But Wolpert et al. argue that if we want to understand the energy costs of real computational systems, relying on Landauer’s bound alone is almost useless. The reason is direct: Landauer’s bound describes an idealized lower limit, whereas real computers never operate without constraints. Real systems must complete computations in finite time, implement state transitions using finite degrees of freedom, resist noise, maintain precision, and often rely on modular architectures, hierarchical organization, and global clocks. All of these constraints introduce additional irreversible entropy production, and these extra costs are often far larger than Landauer’s bound itself.

The core argument of the paper can be summarized as follows: the main energetic cost of computation does not come from abstract information transformation, but from the reliable physical implementation of that transformation under constraints.

Wolpert et al. provide a more rigorous thermodynamic language for this problem.

The paper first reviews the concept of resources in theoretical computer science. Traditional complexity theory focuses on resources such as time, space, communication rounds, and number of gates. A Turing machine has running steps and tape space; a Boolean circuit has gate count and circuit depth; a distributed computing system has message number and communication rounds. These resources matter because they approximate the operating costs of real computers. Yet theoretical computer science has long neglected the most physical resource of all: energy.

This is not accidental. Traditional computation theory abstracts computers as symbol-transforming devices, not as thermodynamic systems. What the input is, what the output is, and how many steps an algorithm requires can all be discussed at an abstract level. But once we ask how these steps are implemented in a real physical system, the question changes. A logic gate is not an abstract function; it is a physical process composed of transistors, currents, voltages, thermal noise, and material constraints. A bit is not a pure symbol, but a distinguishable physical state. An algorithm is not suspended in mathematical space, but realized as a sequence of state transitions constrained by time, precision, and noise.

The authors therefore argue that computational complexity must be integrated with nonequilibrium statistical physics. This is precisely where stochastic thermodynamics becomes relevant. It allows us to analyze physical systems that are far from thermal equilibrium, evolve in finite time, and contain many heterogeneous degrees of freedom. Real computers are exactly such systems. They are not quasistatic processes, nor small perturbations around equilibrium. They are nonequilibrium systems that continuously consume energy, rapidly change states, and exchange heat with thermal reservoirs.

A basic formula in stochastic thermodynamics is entropy production:

$\Sigma = \Delta S + \sum_i \frac{Q_i}{k_BT_i} \geq 0.$

Here, $\Delta S$ is the change in Shannon entropy of the system’s state distribution, $Q_i$ is the heat released from the system to the $i$-th heat reservoir, and $T_i$ is the temperature of that reservoir. Landauer’s bound can be viewed as the lower bound given by this formula under certain idealized conditions. But the paper emphasizes that in real computation, $\Delta S$ is often small, and may even be close to zero in steady-state systems or periodically operating circuits. What dominates energy consumption is irreversible entropy production: the extra dissipation generated by physical constraints.

This step is crucial. It shifts the “thermodynamic cost of information” from a story about bit erasure to a story about constrained dynamics. The question is no longer simply, “How much energy does it take to erase one bit?” Rather, the question becomes: “How much irreversible entropy must a physical system produce in order to reliably implement a state transformation under finite time, finite noise, and finite degrees of freedom?”

The paper identifies several major sources of this cost.

The first is mismatch cost. A physical process can usually minimize entropy production only for a particular initial distribution. If the actual input distribution differs from this optimal distribution, then even if the same process is thermodynamically reversible under the designed distribution, it will produce extra entropy under the actual input distribution. The paper expresses this mismatch cost using KL divergence: if the difference between the actual initial distribution and the optimal initial distribution is not fully preserved at the output, that difference is converted into additional dissipation.

In other words, a system does not compute over an abstract input space. It operates under a particular real input distribution. If the statistical structure of the external world does not match the system’s design assumptions, the system must pay an additional cost to maintain reliability. Energy efficiency is therefore not merely a property of hardware; it is also a result of the match between the system and the distribution of its environment.

The second source is the thermodynamic speed limit. Real computation must be completed in finite time. The faster a computation is performed, the faster the system’s probability distribution must move through state space. Rapidly changing physical states requires additional dissipation. Stochastic thermodynamics provides speed limit theorems that lower-bound entropy production: if a system must move from an initial distribution to a final distribution in a shorter time, while the available physical activity is limited, the minimum entropy production increases.

This means that speed is not free. In traditional computational complexity, “time complexity” mainly counts the number of operations. Stochastic thermodynamics reminds us that physical time itself is a thermodynamic constraint. A fast system is not simply a slow system compressed into a shorter duration. It must dissipate additional energy to ensure that state transitions occur quickly enough.

The third source is the thermodynamic uncertainty relation. If a system requires higher statistical precision, it must generate more entropy. Intuitively, digital computation demands logical states that are sharply separated, voltage curves that are steep, and switching behavior that is reliable. These requirements all amount to suppressing fluctuations. Suppressing fluctuations is not an abstract operation; it is a physical, energy-consuming process. Thus, higher precision requires higher energetic cost.

This creates an interesting contrast with neural systems. Digital computation tends to make states extremely discrete, stable, and precise in order to eliminate noise. Biological systems, in contrast, often work within noise, using population coding, redundancy, probabilistic representation, and closed-loop feedback to exploit uncertainty. The paper does not develop this point in detail, but it provides a basic framework: precision, speed, and energy consumption are inevitably coupled.

The fourth source is the cost of periodic processes. Modern digital computers are usually driven by a global clock, with the same physical process repeated at each clock cycle. On the surface, periodic design simplifies engineering. But the paper points out that if the system’s state distribution at the beginning of each cycle is not in the corresponding periodic steady state, then repeating the same process will inevitably generate entropy production. In other words, the engineering convenience of a global clock may itself carry a thermodynamic cost.

This is also useful for thinking about biological computation. The brain is not a synchronous digital system driven by a global clock. It is better understood as a system composed of multiscale rhythms, local events, asynchronous communication, and state-dependent dynamics. Such a system is not necessarily more “efficient” for every task, but it may avoid some unnecessary dissipation caused by forced global synchronization.

The paper also discusses the distinction between logical reversibility and thermodynamic reversibility. Logical reversibility concerns whether a mapping between states is one-to-one. Thermodynamic reversibility concerns whether the evolution of probability distributions produces entropy. These are not the same thing. A logically reversible process can still dissipate heat in its physical implementation. A single run of a computation may appear not to erase information, but the full cycle—from input generation to computation, output production, and then overwriting the previous input with a new one—still generates unavoidable thermodynamic costs.

This prevents a common misunderstanding: that designing logically reversible computation is sufficient to eliminate energy consumption. The paper’s position is more cautious. Logical structure is only one part of the constraint. The real energetic cost depends on the full physical process, including input distributions, timing control, state initialization, noise suppression, communication, and cyclic operation.

The paper then discusses stochastic thermodynamics in artificial and biological computation.

In artificial systems, researchers have begun using stochastic thermodynamics to analyze low-voltage electronic circuits, CMOS devices, subthreshold transistors, stochastic computing, and probabilistic bits. One particularly interesting observation is that current digital computers spend energy resisting thermal fluctuations in order to maintain determinism. Yet many computational tasks themselves require randomness, such as Monte Carlo sampling, probabilistic inference, and stochastic optimization. The system therefore first spends energy suppressing natural stochasticity, and then uses algorithms to artificially generate pseudorandomness. This may not be physically efficient. Stochastic computing instead aims to use controllable thermal fluctuations directly, rather than treating fluctuations as errors from the start.

In this sense, noise should not only be regarded as a disturbance to be eliminated. The better question is: under what interface, task, and constraint can noise become usable structure? Fluctuation itself is not information, but under suitable dynamics and selection constraints, fluctuation can participate in search, sampling, representation, and learning.

In biological systems, the paper begins with kinetic proofreading. Protein synthesis and molecular recognition require extremely high accuracy, which cannot be achieved through a single-step reversible process. Hopfield’s theory of kinetic proofreading shows that biological systems can greatly reduce error rates by consuming energy to introduce irreversible steps. Here, energy is not simply wasted; it is used to buy precision.

More generally, cells must sense the environment, transmit information, and regulate responses. Learning about the external environment requires breaking detailed balance and consuming energy. Transferring information between spatially separated cellular components also consumes a substantial amount of energy. At the neural level, the brain must sense, represent, and transmit information across large populations of neurons. The paper notes that the human brain accounts for about 2% of body mass but consumes about 20% of metabolic energy. Understanding how the brain represents and transmits information while minimizing energy consumption remains a fundamental problem in neuroscience.

This complements the previous discussion on self-powered analog neuromorphic systems. Kim et al. emphasize that environmental stimuli can serve simultaneously as information sources and energy sources, allowing a system to sense, encode, and learn directly through a physical interface. Wolpert et al., in contrast, emphasize that even after information has entered the system, if the system needs to use that information reliably under finite time, finite noise, finite precision, and finite structural constraints, it must still pay a thermodynamic cost. The former shows how external structure can help a system save energy; the latter shows how internal constraints determine how far this energy saving can go.

Therefore, this paper does not contradict the earlier idea of the decoupling between information complexity and energy cost. Rather, it supplies a stricter boundary for that idea. The structure in the world can be highly complex, but the system does not need to pay for the existence of that structure itself. It only needs to pay for accessing, distinguishing, stabilizing, and using that structure. Stochastic thermodynamics further shows that these costs arise from concrete constraints: speed, precision, input-distribution mismatch, periodicity, modularity, communication, and physical implementation.

In the second half of the paper, the authors identify several open research directions. Among the most important are modularity, hierarchy, and communication cost.

Real computational systems, whether artificial or biological, are usually not fully connected, homogeneous, centrally controlled wholes. They are modular and hierarchical. Modularity can improve robustness and reduce global coupling in complex systems. Hierarchy can organize multiscale processes and reduce the difficulty of managing long-range correlations. But these structures may also introduce additional entropy production. The paper notes that modularity is known to increase entropy production, even though it is often valuable for robustness and resource utilization. How the benefits of modular and hierarchical organization balance against their thermodynamic costs remains poorly understood.

This is especially important for understanding the brain. The brain is clearly modular and hierarchical: sensory systems, motor systems, hippocampus, cortex, thalamus, basal ganglia, and other structures have distinct functions and connectivity patterns. Communication among them is not free. Neural energy efficiency cannot come only from the fact that single neurons are energy-efficient. It must also depend on how the overall architecture constrains communication, supports local processing, triggers events, and shares states.

The paper also emphasizes communication cost. Computation includes not only transforming information, but also transferring information. In modern computers, information transfer among components is one of the major sources of heat. In biological systems, long-range communication is also expensive. Research on entropy production in communication systems is still in its early stages, but it may become central to understanding the energy cost of real computation.

This connects to my earlier discussion of Shannon and Weaver’s three levels of communication. Traditional information theory mainly solves Level A: how symbols can be reliably transmitted. Intelligent systems must also confront Level B and Level C: which structures have meaning, and which meanings can guide behavior. Wolpert et al. add another physical dimension: even at Level A, reliable transmission is not free. Once we move into Level B and Level C, the system must also decide which information is worth transmitting, where it should be transmitted, with what precision, and on what timescale. These choices are themselves constraint problems involving energy, information, and behavior.

Thus, the energetic cost of computation should not be understood as “the more complex the information, the higher the energy cost.” A more precise statement is: energy cost appears when a system attempts to stabilize certain external or internal differences as operable states, and to transmit, maintain, and use those states for behavior within finite time. Information complexity itself is not the direct cost. Accessibility, distinguishability, reliability, speed, and communication are the main sources of cost.

This also explains why the energy efficiency of the brain cannot be explained simply by saying that neurons are more efficient than transistors. The brain certainly consumes substantial energy, and far more than the abstract thermodynamic lower bound. But it may be efficient at another level. It does not attempt to turn all information into globally precise digital states. It does not synchronize all computation to a global clock. It does not transmit all raw data to a central processor. Instead, through the body, receptors, local circuits, rhythms, synaptic plasticity, and behavioral feedback, it constructs a multilayered set of interfaces that allow only those differences relevant to the current state and task to enter usable dynamics.

In other words, the brain may be energy-efficient not because it violates thermodynamics, but because it respects thermodynamics. It avoids unnecessary global precision, reduces unnecessary communication, exploits input statistics, allows local noise to exist, and continuously changes its own input distribution through closed-loop behavior. It does not compute the world from outside the world; it regulates its coupling with the world from within it.

What is most valuable in this paper is that it connects the abstract problem of information with real physical processes through the lens of energy cost. Computation is not the symbolic mapping itself. It is the transformation of the distribution over information-bearing degrees of freedom in a physical system under constraints. Wherever there are constraints, there is irreversible entropy production. Wherever speed, precision, and reliability are required, energy must be paid. Wherever there are modules and communication, structural dissipation must be faced.

We can therefore extract a clearer formulation from this paper: the existence of information does not need to be paid for by the system, but the usability of information does. A system does not consume energy for the structure already present in the world. It consumes energy to maintain the physical interfaces that can access, distinguish, stabilize, transmit, and use that structure. Stochastic thermodynamics provides the mathematical language for describing these costs.

If the previous article on self-powered analog neuromorphic systems showed how environmental stimuli can serve simultaneously as information sources and energy sources, Wolpert et al.’s paper clarifies the other side of the problem. Even when the environment already provides structure and physical driving force, a system that turns that structure into reliable computation must still pay an entropy-production cost determined by physical constraints. True energy efficiency is not the elimination of energy cost. It is the ability to spend energy on necessary interfaces while avoiding wasteful precision, synchronization, communication, and reconstruction.