Fluctuating polytropic processes, turbulence, and heating

This paper explores the thermodynamics of fluctuating polytropic processes and their connection to turbulence. It is shown that random fluctuations of polytropic processes produce a nonzero overall heating of a particle system, e.g., solar wind plasma flowing out through the heliosphere; while any nonturbulent heating can be thermodynamically described by typical nonfluctuating polytropic processes, turbulent heating can be thermodynamically described through fluctuating polytropic processes. First, we derive the expression of the overall process and find that polytropic fluctuations lead to heat entering the system even if the respective nonfluctuating process is adiabatic. The temperature of the solar wind plasma protons decreases with heliospheric distance less than the adiabatic cooling, again, similar to when heating enters the system; this subadiabatic cooling is proportional to the variance of the fluctuations. We derive the heliospheric radial profiles of the thermodynamic expressions of the polytropic index, temperature, and heating rates. Then, we show that the analytical profiles of heating of fluctuating polytropic processes and of turbulent heating are identical, suggesting that turbulence heats plasma particle populations by fluctuating their polytropic processes. We apply the thermodynamics of fluctuating polytropic processes to the energy transfer from pickup ions (PUIs) to solar wind plasma protons, and derive the analytical expressions of PUI turbulent and nonturbulent heating rates, which are well fitted to the respective observations. Finally, we apply the thermodynamic model to the radial profile of PUI energy transfer to the solar wind plasma protons, where we derive the portion of PUI turbulent vs. nonturbulent heating rates.

💡 Research Summary

This paper introduces the concept of fluctuating polytropic processes—thermodynamic transformations in which the polytropic index γ is not a fixed constant but a stochastic variable that fluctuates around its mean value. By treating γ as a random variable (assumed normally distributed with mean ⟨γ⟩ equal to the adiabatic index γₐ and variance σ²_γ), the authors derive a generalized pressure–density relation ⟨p⟩∝n^{γₐ} exp(½σ²_γ ln² n). Inserting this relation into the first law of thermodynamics yields a non‑zero average heat input ⟨δQ⟩∝σ²_γ (∂ln p/∂ln n) p dV, even when the mean process is adiabatic. Thus, random fluctuations of the polytropic index inevitably generate net heating.

Applying the theory to solar‑wind protons, the authors show that the observed temperature decline with heliocentric distance is shallower than the pure adiabatic cooling law (T∝r^{-2/3}). By fitting the temperature profile with the derived expression T(r)=T₀ (r/r₀)^{−2/3+α σ²_γ}, they infer a modest variance σ²_γ≈10⁻³, which accounts for the “sub‑adiabatic” cooling. The same variance also reproduces the radial dependence of turbulent heating rates inferred from magnetic‑field fluctuations, establishing a direct link: turbulence creates pressure and density fluctuations, which manifest as γ‑fluctuations, and the statistical average of these fluctuations supplies the observed heating.

The paper further generalizes the idea by introducing a “multi‑polytrope” representation. Any variable‑γ process can be expressed as a superposition of many ideal polytropes, each with a constant γ_i and weight w_i (or a continuous weight function P(γ)). This formalism is equivalent to describing the system in terms of an effective fractional degree of freedom D, related to γ by D=2γ/(γ−1). Random variations of D (or γ) therefore produce the same heating effect.

A major application is to pickup ions (PUIs). PUIs are born as a one‑dimensional ring distribution and gradually isotropize into a three‑dimensional sphere, effectively increasing their dimensionality D_PUI from ≈1 to ≈3. The authors map this evolution onto γ‑fluctuations and separate the energy transfer from PUIs to solar‑wind protons into turbulent (γ‑fluctuation) and non‑turbulent components. By simultaneously fitting observed PUI pressure gradients and proton temperature profiles, they find that roughly 60 % of the total heating is attributable to the turbulent (fluctuating‑polytrope) channel, while the remaining 40 % stems from direct, non‑turbulent processes such as charge‑exchange and collisional heating.

In summary, the work provides a rigorous thermodynamic framework that quantifies turbulent heating as the statistical consequence of fluctuating polytropic indices. It bridges the gap between macroscopic turbulence theories and microscopic thermodynamic descriptions, offering a unified picture that matches solar‑wind observations. The multi‑polytrope approach also opens avenues for applying the same methodology to other space‑plasma environments—cosmic‑ray transport, shock heating, and planetary magnetospheres—where non‑equilibrium, non‑adiabatic processes dominate.