[Paper Review] Quantum-mechanical engine models and their efficiencies

Jianhui Wang, Yongli Ma+1 more|arXiv (Cornell University)|Feb 3, 2013

Advanced Thermodynamics and Statistical Mechanics1 references3 citations

TL;DR

This paper proposes a quantum-mechanical extension of classical heat engine cycles—Carnot, Brayton, Otto, and Diesel—by redefining thermodynamic processes using quantum systems with well-defined energy spectra rather than temperature. It derives the efficiency of each quantum cycle, showing that the efficiency expressions are identical to their classical counterparts, thus establishing the universality of efficiency across classical and quantum regimes for a given cycle type.

ABSTRACT

Based on quantum thermodynamic processes, we make a quantum-mechanical (QM) extension of the typical heat engine cycles, such as the Carnot, Brayton, Otto, and Diesel cycles, etc. The temperature is not included in these QM engine cycles, as lies in the fact that the concept of energy is well-defined in quantum mechanics, temperature \emph{a priori} is not. These QM engine cycles are implemented by an ideal or interacting system with an arbitrary number of particles confined in an arbitrary power-law trap. As a result, a relation between the quantum adiabatic exponent and trap exponent is found. The efficiency of a given QM engine cycle is similar to that of its classical counterpart, thereby identifying the universality of the efficiency.

Motivation & Objective

• To extend classical heat engine cycles (Carnot, Brayton, Otto, Diesel) into the quantum regime using energy-based thermodynamics instead of temperature.
• To define quantum analogs of classical thermodynamic processes—isoenergetic, isochoric, isobaric, and adiabatic—based on quantum state evolution and energy spectrum control.
• To establish a relation between the quantum adiabatic exponent and the trap potential exponent in systems confined in power-law traps.
• To demonstrate that the efficiency of a given quantum engine cycle is independent of system-specific parameters and matches its classical counterpart, proving universality of efficiency.

Proposed method

• Model a quantum system of N particles in a power-law trap with single-particle energy spectrum εₙ = ℏωn^σ, where ω ∝ L^(-θ), and θ is the trap exponent.
• Define quantum thermodynamic processes via control of the system's Hamiltonian: isoenergetic (constant energy expectation), isochoric (constant L), isobaric (constant pressure F), and adiabatic (no energy exchange with bath).
• Use the first law of quantum thermodynamics: dE = δQ + δW, with δQ = Σεₙdpₙ and δW = Σpₙdεₙ, to describe energy and work exchange.
• Derive the force on the trap wall as F = θΣpₙεₙ/L, linking pressure to energy and trap size, and express internal energy as E = FL/θ.
• Construct quantum cycles (e.g., Otto: two isochoric and two adiabatic processes) and compute efficiency using energy and pressure changes at key instants.
• Use the relation between trap exponent θ and quantum adiabatic exponent γ = θ + 1 to express efficiencies in terms of geometric parameters like L_c and L_h.

Experimental results

Research questions

• RQ1 Can classical heat engine cycles be consistently extended into the quantum regime without relying on temperature?
• RQ2 What are the quantum analogs of classical thermodynamic processes such as isochoric, isobaric, and adiabatic processes?
• RQ3 How does the efficiency of a quantum heat engine cycle compare to its classical counterpart?
• RQ4 Is there a universal relation between the quantum adiabatic exponent and the trap potential exponent in quantum systems?
• RQ5 Does the efficiency of a given quantum engine cycle depend on system-specific parameters like particle number or interaction strength?

Key findings

• The efficiency of the quantum Otto cycle is η_O = 1 - (L_c / L_h)^(θ) = 1 - (L_c / L_h)^(γ-1), identical to the classical Otto engine efficiency.
• The efficiency of the quantum Diesel cycle is η_D = 1 - (1/(θ+1)) × [(L₂^θ⁺¹ - L₁^θ⁺¹)/((L₂ - L₁)L₃^θ)], matching the classical Diesel expression.
• The quantum Ericsson and Stirling cycles achieve the same efficiency as the quantum Carnot cycle: η = 1 - E_c / E_h, confirming universality.
• A relation between the quantum adiabatic exponent γ and the trap exponent θ is derived as γ = θ + 1, linking trap geometry to thermodynamic response.
• The efficiency expressions are independent of particle number, interaction strength, and specific system parameters, proving universality across diverse quantum systems.
• The results hold for arbitrary systems, including ideal or interacting particles in power-law traps, harmonic systems, spin-1/2 systems, and cavity radiation fields.