Statistics of power and efficiency for collisional Brownian engines

TL;DR

The study investigates joint fluctuations of power and efficiency in collisional Brownian engines within stochastic thermodynamics, showing that the full statistics are determined by Onsager coefficients and that fluctuations can realize events surpassing mean-optimal values. It derives Gaussian distributions for stochastic powers and a Cauchy-like form for stochastic efficiency, and provides analytic expressions for conditional distributions under power constraints. By analyzing both marginal and conditional statistics, the work reveals how rare, high-power fluctuations can be accompanied by high efficiency and how conditioning on power shapes the likelihood of favorable efficiency outcomes. The framework, robust to protocol details, offers a general approach to understanding fluctuation-driven performance in microscopic engines and can be extended to broader classes of thermal machines.

Abstract

Collisional Brownian engines have attracted significant attention due to their simplicity, experimental accessibility, and amenability to exact analytical solutions. While previous research has predominantly focused on optimizing mean values of power and efficiency, the joint statistical properties of these performance metrics remain largely unexplored. Using stochastic thermodynamics, we investigate the joint probability distributions of power and efficiency for collisional Brownian engines, revealing how thermodynamic fluctuations influence the probability of observing values exceeding their respective mean maxima. Our conditional probability analysis demonstrates that when power fluctuates above its maximum mean value, the probability of achieving high efficiency increases substantially, suggesting fluctuation regimes where the classical power-efficiency trade-off can be probabilistically overcome. Notably, our framework extends to a broader class of engines, as the essential features of the statistics of the system are fully determined by the Onsager coefficients. Our results contribute to a deeper understanding of the role of fluctuations in Brownian engines, highlighting how stochastic behavior can enable performance beyond traditional thermodynamic bounds.

Figures (6)

• Figure 1: Depiction of the efficiency ${\bar{\eta}}$ (top) and mean power $\bar{\dot{W}}_2$ (bottom) versus different $X_2=Tf_2$ for different $\tau$'s. Green curves show results for $\tau_1=0.1$, with $f_2^{\rm ME}=-0.972$, $\eta^{\rm ME}=0.944$, $\dot{W}_2^{\rm ME}=0.0068$ at maximum efficiency, and $f_2^{\rm MP}=-0.500$, $\eta^{\rm MP}=0.499$, $\dot{W}_2^{\rm MP}=0.0625$ at maximum power. Pink curves show results for $\tau_2=1$, with $f_2^{\rm ME}=-0.750$, $\eta^{\rm ME}=0.563$, $\dot{W}_2^{\rm ME}=0.0401$ at maximum efficiency, and $f_2^{\rm MP}=-0.480$, $\eta^{\rm MP}=0.428$, $\dot{W}_2^{\rm MP}=0.0588$ at maximum power. Symbols denote the corresponding maximum values $\bar{\dot{W}}_2^{MP}$ and $\eta_{ME}$.
• Figure 2: Probability distribution of power ${\dot W}_2$. (a) Probability that power exceeds its maximum mean value, which is maximized at the force corresponding to the maximum average power. (b) Probability that power falls within an intermediate range, which is maximized at a distinct value $f_2^{\rm{INT}}$. For $\tau_1 = 0.1$ and $\tau_2 = 1$, the values of $f_2^{INT}$ are approximately $-0.0126$ and $-0.0583$, yielding probabilities of $43\%$ and $12\%$, respectively.
• Figure 3: Probabilities for the efficiency. a) Probability to have the efficiency being greater than the efficiency at maximum mean power. For both time intervals, the maximum probability is 50%. b) Probability to have the efficiency being greater than the mean efficiency at maximum power, but less than the mean maximum efficiency. The optimal force values maximizing this probability are $f_2^{\mathrm{INT}} = -0.686$ for $\tau_1 = 0.1$ and $f_2^{\mathrm{INT}} = -0.492$ for $\tau_2 = 1$, yielding probabilities of approximately $90\%$ and $20\%$, respectively.
• Figure 4: Probability distributions. a) Probability distribution for the efficiency. b) Conditional probability distribution for the efficiency with the condition of $\dot{w}_2 \in \Omega_1 = [\bar{\dot{W}}_2^{ME},\bar{\dot{W}}_2^{MP}]$. c) Conditional probability distribution for the efficiency with the condition of $\dot{w}_2 \in \Omega_1 = [\bar{\dot{W}}_2^{MP},\infty)$. All the distributions are compared for two thermodynamic force values, $f_2^{MP}$ and $f_2^{ME}$. For all plots, we choose $\tau=\tau_2=1$.
• Figure 5: Conditional probability, with $\Omega_1 = [\dot W_2^{\rm ME},\dot W_2^{\rm MP}]$. a) Conditional Probability for the efficiency being greater than the average maximum efficiency. For $\tau_1 = 0.1$, the optimal force is $f_2^{\mathrm{C1M}} = -0.974$ with maximum probability of $\sim 8\%$; for $\tau_2 = 1$, the optimal force is $f_2^{\mathrm{C1M}} = -0.963$ with maximum probability of $\sim 1.2\%$. b) Conditional Probability for the efficiency being in the intermediary range. For $\tau_1 = 0.1$, the optimal force is $f_2^{\mathrm{C1I}} = -0.654$ with maximum probability of $\sim 20\%$; for $\tau_2 = 1$, the optimal force is $f_2^{\mathrm{C1I}} = -0.577$ with maximum probability of $\sim 5\%$. For all cases, the forces that maximize these probabilities differ from the reference forces.