We derive ignorance based prior distribution to quantify incomplete information and show its use to estimate the optimal work characteristics of a heat engine.Subjective probability or Bayesian inference methods seek to quantify uncertainty due to incomplete prior knowledge about the system [1,2]. The incomplete information is quantified as a prior probability distribution, or simply known as a prior and interpreted in the sense of degree of belief about the likely values of the uncertain parameter. The choice of an appropriate prior has been a crucial issue in the Bayesian epistemology, which has hampered its development and general acceptance for a long time. To outline the framework, consider a system which depends on two parameters T 1 and T 2 , where each parameter may lie within a specified range, [T − , T + ]. Further, we are told that the values of T 1 and T 2 are constrained by a known relation, T 1 = F (T 2 ), so that given a value for one parameter, it implies a certain value of the other. Assume now that we have incomplete information about the system, which refers to an ignorance about the exact values of the above parameters. In this paper, we present evidence that inference based on subjective ignorance, has a close relation with optimal characteristics of certain thermodynamic systems. The present approach was recently proposed and
When incomplete information about the control parameters is quantified as a prior distribution, a subtle connection emerges between quantum heat engines and their classical analogs. We study the quantum model where the uncertain parameters are the intrinsic energy scales and compare with the classical models where the intermediate temperature is the uncertain parameter. The prior distribution quantifying the incomplete information has the form π(x) ∝ 1/x in both the quantum and the classical models. The expected efficiency calculated in near-equilibrium limit approaches the value of one third of Carnot efficiency.
We consider the standard thermodynamic processes with constraints, but with additional uncertainty about the control parameters. Motivated by inductive reasoning, we assign prior distribution that provides a rational guess about likely values of the uncertain parameters. The priors are derived explicitly for both the entropy conserving and the energy conserving processes. The proposed form is useful when the constraint equation cannot be treated analytically. The inference is performed using spin-1/2 systems as models for heat reservoirs. Analytical results are derived in the high temperatures limit. Comparisons are found between the estimates of thermal quantities and the optimal values described by extremum principles. We also seek a intuitive interpretation of the prior and show that it becomes uniform over the quantity which is conserved in the process. We find further points of correspondence between the inference based approach and the thermodynamic framework.
Abstract:The thermal characteristics of a heat cycle are studied from a Bayesian approach. In this approach, we assign a certain prior probability distribution to an uncertain parameter of the system. Based on that prior, we study the expected behaviour of the system and it has been found that even in the absence of complete information, we obtain thermodynamic-like behaviour of the system. Two models of heat cycles, the quantum Otto cycle and the classical Otto cycle are studied from this perspective. Various expressions for thermal efficiences can be obtained with a generalised prior of the form Π( ) ∝ 1/ . The predicted thermodynamic behaviour suggests a connection between prior information about the system and thermodynamic features of the system.
PACS
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