A fuelled model, not a tested machine
A mathematical study has derived an exact picture of how chemical fuel, heat and mechanical load interact in a small Brownian engine. In the model, a particle moves around a three-state loop—0 to E to 2E and back to 0—touching two cold links and one hot link. A fuel molecule is consumed on the hot transition. The results are equations for this network, not measurements from a real motor.
The preprint derives stationary probabilities, the cycle current—the average directed traffic around the loop—and heat flows, entropy production, exergy balance, operating modes and optimization conditions. It uses local detailed balance, a rule that keeps each modeled transition consistent with its energy and temperature. The supplied manuscript is an arXiv preprint, version v1, dated 26 Aug 2026.
Stall is a balance of heat, fuel and load
The central quantity is cycle affinity, a dimensionless measure of the net push around the loop. The analysis writes it as A = E(1/Tc − 1/Th) + Δµ/Th − f(2/Tc + 1/Th), where E is the model’s energy step, Tc and Th are the cold and hot temperatures, Δµ is the chemical free-energy drop, and f is the mechanical load.
The engine stalls when A = 0, giving the exact load fstall = [E(Th − Tc) + TcΔµ]/(2Th + Tc). The formula puts thermal and chemical driving in the same balance: changing the fuel input changes the mechanical load at the zero-current boundary.
Under the model’s unicyclic assumptions, stall is reversible. The current falls to zero, detailed balance is restored, and entropy production is zero.
One especially sharp cancellation occurs at the compensation value Δµ* = 3E/2. For every Th greater than Tc, the stall load is then E/2, and both reversible cycle heats vanish. The result is an energetic cancellation within the model, not the removal of the temperature difference between the hot and cold reservoirs.
More fuel raises current—but only to a ceiling
Chemical driving does not increase modeled speed without limit. At fixed mechanical parameters, stationary current rises strictly as Δµ increases in both Metropolis branches, but approaches a finite kinetic ceiling. In the limiting picture, stronger fuel drive removes only one of the loop’s three kinetic bottlenecks, leaving the others to cap throughput.
The full analysis also reports a lower speed-optimal thermal barrier at larger Δµ, while motion persists when that thermal barrier is very small. These finite-current and barrier results belong to the chosen Metropolis kinetics; the authors note that finite current, power and barrier optima can depend on the kinetic convention, even though affinity and stall are more robust across rate families that satisfy local detailed balance.
The engine changes character across the map
Heat flow divides the forward operating map into three regimes. In a hybrid heat-engine regime, the signs are qh > 0 and qc > 0. In a chemical-motor regime, fuel-driven cycling dissipates heat into both baths, with qh < 0 and qc > 0. In a third regime, the chemical motor refrigerates the cold bath while sending heat into the hot bath, with qh < 0 and qc < 0.
Those regimes show why a single label such as “heat engine” would be incomplete: chemical free energy can drive the cycle while the baths absorb heat, or while one bath is cooled. The model’s exergy balance treats chemical free energy as a fully available resource, whereas heat contributes only the fraction that can be converted under the Carnot limit.
Power has an exact condition, with a narrower half-stall result
Power optimization is not reduced to a universal half-stall rule. The model supplies an exact scalar stationarity equation for the maximum-power load; only close to equilibrium does it simplify to fmp = fstall/2. In that same near-equilibrium limit, the stated exergy efficiency is 3/(6 − ηC), where ηC is the Carnot efficiency.
The reversible exergy identity behind that result is W = Δµ + ηCqh. In that accounting, mechanical work is decomposed into chemical free energy plus the Carnot-convertible part of hot heat. It is a balance for the model’s reversible limit, not a claim that all supplied heat becomes work.
Where the chemical fuel is attached matters
A broader unicyclic result concerns the temperature of the chemically gated transition—the link where fuel is coupled. Chemical leverage at stall scales with the inverse of that local temperature, written as a factor proportional to 1/Tch. In practical terms, the same chemical free-energy input has a different effect on stall depending on which thermal link carries it.
Stopping motion does not always stop dissipation
The clean reversibility at stall changes when the network has more than one cycle. In the minimal multicyclic extension, the model allows zero mechanical velocity, V = 0, while an attached futile chemical loop continues to run. Total entropy production remains positive, expressed as Ṡtot = JfAf > 0. Fuel can therefore be consumed and dissipated even when the mechanical coordinate shows no net motion.
That result marks a change in network structure rather than a contradiction of the one-cycle result. In the three-state unicyclic model, zero affinity at stall goes with zero current and zero entropy production; in the extended network, mechanical velocity can cancel while an internal cycle retains positive dissipation.
The model’s boundaries
The authors present the framework as a bridge between Brownian heat engines and chemical molecular motors, but the supplied evidence does not validate a real device. No biological or physical measurements were analyzed; the evidence consists of exact derivations, asymptotic analyses and numerical illustrations for the specified models.
The primary model fixes a three-state topology, a particular energy sequence, one chemically coupled hot event and Metropolis rates with a common attempt frequency. The authors note that the half-stall maximum-power result is a near-equilibrium approximation, while behavior in more general interacting multicyclic networks remains to be tested.
All results are analytical. Scripts used to evaluate the exact expressions and generate the figures are available from the author upon reasonable request. The acknowledgments report support from the Science Division of West Los Angeles College, with no grant details provided.
Paper data and sources
Original title: Exact chemo--thermal Metropolis Brownian engine: chemical leverage, temperature-neutral stall, power optimization, and multicyclic dissipation
Authors: Mesfin Taye
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text