Climate is a system far from thermal equilibrium: it receives energy from the Sun at about 6000 K and radiates it into space at about 3 K. Under such conditions, the familiar statistical mechanics of Boltzmann and Gibbs, based on detailed balance, does not apply. Instead, a nonequilibrium steady state (NESS) emerges, featuring persistent probability fluxes that form closed loops in phase space. These loops are invisible in equilibrium, yet they may explain observed climate oscillations such as El Niño and the Madden–Julian oscillation. This work stands apart: it has few close neighbors, with the nearest studies touching on quantum states, instantons, and quantum entanglement, which are quite different areas. Thus, this paper fills a gap by linking abstract concepts of nonequilibrium statistical physics to concrete climate data.
The authors use a linear Gaussian model (LGM), the simplest type of stochastic differential equation with additive noise. Such a model is fully specified by two matrices: the relaxation matrix A and the diffusion matrix D. From observational data, they construct covariance matrices C0 (simultaneous) and Cτ (with time lag τ), then compute A from the relation Cτ = $e^{Aτ}$ C0, and D from the generalized Einstein relation AC0 + C0A^T + 2D = 0. For El Niño, they use monthly data of the NINO3 index and the depth of the 20°C isotherm for 1960–2016; for the Madden–Julian oscillation, daily OMI index data for 1979–2016. The probabilistic angular momentum L(t) is computed as the antisymmetric part of the tensor product x(t) ∧ x(t+ε)/ε, where ε is the time step. Then a histogram of L(t) is built and compared with the theoretical distribution derived from the LGM.
For both climate oscillations, the histograms of probabilistic angular momentum turned out to be strongly asymmetric, with exponential tails. The mean ⟨L⟩ is nonzero, although the most probable value (mode) is zero. This indicates time-reversal symmetry breaking and hence the nonequilibrium character of the system. For example, for El Niño, the mean probabilistic angular momentum is positive, corresponding to rotation in the phase space of the NINO3 and d20 indices. The distributions predicted by the linear Gaussian model agree well with the observed distributions, despite the complexity of real processes. The key relation ⟨L⟩/σ_ℓ = sqrt(ℓ²/(σ₀²+ℓ²)) shows that the mean cannot exceed the standard deviation, so drawing meaningful conclusions requires analyzing the full distribution, not just the mean.
This work stands out from its neighbors by applying the concept of probabilistic angular momentum to a macroscopic climate system rather than to microscopic quantum or classical models. Nearby studies in the archive concern quantum measurements, instantons, and quantum entanglement, which operate on a completely different scale and subject. Here, for the first time, a practical method is proposed to extract information about nonequilibrium currents from real climate data. This opens the way to using probabilistic angular momentum as a universal diagnostic tool for any nonequilibrium steady system, not just climate.
Probabilistic angular momentum can be applied to other climate phenomena: the North Atlantic Oscillation, the Southern and Northern Annular Modes, and the variability of the Atlantic Meridional Overturning Circulation. Also, the behavior of the time correlation function ⟨x(0) ∧ x(τ)⟩* as a function of τ remains to be explored: preliminary data show that it first rises to a maximum and then decays exponentially. The physics of this peak is not yet clear. Furthermore, it is important to determine which variables are 'leaders' and which are 'followers' in nonequilibrium systems, and whether a more general theory of nonequilibrium fluctuations can be built on the basis of probabilistic angular momentum.
The new tool will improve the diagnosis and comparison of climate models, and allow checking how well models reproduce nonequilibrium currents that are important for long-term predictability. More broadly, the method applies to any nonequilibrium steady system, from biological to socioeconomic ones.
Apply the probabilistic angular momentum analysis to other climate indices and to climate model data to assess their ability to reproduce nonequilibrium currents. Develop a theoretical understanding of the behavior of the time correlation function and its peak.
In statistical physics, an open question remains: how to generalize the concepts of energy and entropy to nonequilibrium steady states. Probabilistic angular momentum could become a key element of such a generalization, as it is directly linked to the breakdown of detailed balance and time reversal. Additionally, the problem of identifying leading and following variables in complex nonequilibrium systems remains unsolved, which is important for understanding causal relationships.
Probabilistic angular momentum has the same units as the diffusion coefficient: length squared divided by time. For El Niño, with its indices of temperature (°C) and depth (cm), the units become exotic: °C·cm/month.
The idea that climate oscillations are manifestations of hidden rotations in an abstract space echoes Isaac Asimov's novel 'The End of Eternity', where changes in reality are described as loops in a multidimensional state space.