Collective dynamics in financial markets can emerge through synchronized movements of large groups of assets. Motivated by analogies with interacting many-body systems, we introduce a spin-lattice representation for analyzing collective states in cryptocurrency markets. In this framework, assets are encoded as binary spin variables according to the sign of their returns, while correlations between assets determine effective interaction strengths. A correlation-based breadth-first search (CBFS) procedure embeds 169 cryptocurrencies into a $13 \times 13$ lattice, enabling the construction of an Ising-like Hamiltonian describing the market configuration, which we call the \emph{Market Crystal}. Macroscopic observables such as magnetization and energy provide a statistical-mechanical characterization of collective market states. The resulting phase-space structure highlights regimes of strong alignment and fragmentation among assets, with an energy--magnetization pattern suggestive of predominantly ferromagnetic interactions. This framework offers a statistical-mechanical viewpoint for studying collective behavior in financial systems.
Klaus M. Frahm, Leonardo Ermann, Dima L. Shepelyansky
According to the recent Wealth Thermalization Hypothesis (WTH) the wealth inequality in the world is described by the Rayleigh-Jeans (RJ) thermal distribution of interacting agents in a society with social stratification. In this concept, the wealth layers of society are associated with energy levels from a nonlinear dynamical system conserving two integrals of motion being total energy and probability norm. This leads to RJ condensation and the formation of a huge poverty phase of low wealth and a tiny oligarchic phase that captures a main part of total society wealth. This RJ phenomenon has similarities with self cleaning in multimode optical fibers and constraint driven condensation in various physical systems. We analyze real Lorenz and Pareto curves for wealth of households in countries and the world, Gross Domestic Product of countries, market capitalization of companies at stock exchange of Hong Kong, Shanghai, London, bitcoin transactions, world trade between countries and show that the WTH theory gives a good description of these curves. On the basis of this comparison we argue that the RJ thermal distribution provides a universal description of wealth inequality in the world.
Jeong-Hyuck Park, Chanhee Park, Claudio J. Tessone, Yu Zhang
Digitalisation transforms money from distinguishable physical objects into fungible informational units. A recent theoretical framework predicts that such indistinguishable wealth obeys bosonic occupancy statistics, leading to geometric ownership distributions and enhanced inequality. Using Bitcoin blockchain data, we test this prediction on 63 UTXO denominations across 72 monthly snapshots (2018--2023). A one-parameter geometric model describes the ownership distributions, reproducing both mean holdings and their temporal evolution; Jensen--Shannon divergence values lie below $0.08$ in $99.74\%$ of cases. The inferred inverse-temperature parameter satisfies the analytic mean--temperature relation to better than $0.1\%$ in every sample -- a self-consistency test that two-parameter alternatives cannot pass -- and remains within a narrow band across eight orders of magnitude in denomination and over six years. Bitcoin UTXO ownership statistics are therefore consistent with bosonic occupancy laws, suggesting that the informational nature of electronic money may act as a structural driver of inequality in digital economies.
Jian Gao, Lufeng Zhang, Ping Fang, Pu Ke · 7 authors
Fluctuation theorems show how coarse graining transforms microscopic symmetry into observable irreversibility. Here we ask whether an analogous symmetrybased diagnostic can be constructed for financial markets. At the microscopic level, each transaction pairs a buyer and a seller, whereas trading decisions are typically made from coarse-grained price histories. Using symmetric takeprofit and stop-loss rules, we compare the holding-time distributions of long and short trading ensembles generated from the same price series. Across equityindices, individual stocks and cryptocurrencies, the log-ratio of the two distributions shows a robust crossover. It remains nearly constant at short durations but becomes linear in the tail, implying an exponential directional asymmetry. The tail slope defines an effective market temperature, an operational measure of fluctuation intensity on the chosen observation scale. A Bachelier first-passage benchmark captures the exponential tails but not the asymmetry, because long and short positions share the same leading decay rate. By contrast, short-time correlations between overlapping positions provide a minimal mechanism for the asymmetry by generating direction-dependent subleading relaxation spectra in a coarse-grained Markov description. Together, these results establish a fluctuation-theorem-like diagnostic of irreversibility in financial markets and, more broadly, in complex systems accessible only through coarse-grained observables.
This paper compares and contrasts stationarity between the conventional stock market and cryptocurrency. The dataset used for the analysis is the intraday price indices of the S&P500 from 1996 to 2023 and the intraday Bitcoin indices from 2019 to 2023, both in USD. We adopt the definition of `wide sense stationary', which constrains the time independence of the first and second moments of a time series. The testing method used in this paper follows the Wiener-Khinchin Theorem, i.e., that for a wide sense stationary process, the power spectral density and the autocorrelation are a Fourier transform pair. We demonstrate that localized stationarity can be achieved by truncating the time series into segments, and for each segment, detrending and normalizing the price return are required. These results show that the S&P500 price return can achieve stationarity for the full 28-year period with a detrending window of 12 months and a constrained normalization window of 10 minutes. With truncated segments, a larger normalization window can be used to establish stationarity, indicating that within the segment the data is more homogeneous. For Bitcoin price return, the segment with higher volatility presents stationarity with a normalization window of 60 minutes, whereas stationarity cannot be established in other segments.
This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.
Modern technology has brought novel types of wealth. In contrast to hard cash, digital currency does not have a physical form. It exists in electronic forms only. To date, it has not been clear what impacts its ongoing growth will have, if any, on wealth distribution. Here, we propose to identify all forms of contemporary wealth into two classes: ‘distinguishable’ or ‘identical’. Traditional tangible moneys are all distinguishable. Financial assets and cryptocurrencies, such as bank deposits and Bitcoin, are boson-like, while non-fungible tokens are fermion - like. We derived their ownership-based distributions in a unified manner. Each class follows essentially the Poisson or the geometric distribution. We contrast their distinct features such as Gini coefficients. Furthermore, aggregating different kinds of wealth corresponds to a weighted convolution where the number of banks matters and Bitcoin follows Bose–Einstein distribution. Our proposal opens a new avenue to understand the deepened inequality in modern economy, which is based on the statistical physics property of wealth rather than the individual ability of owners. We call for verifications with real data.
Pedro D. Manrique, Frank Huo, Sara El Oud, Minzhang Zheng · 6 authors
Online communities featuring 'anti-X' hate and extremism, somehow thrive online despite moderator pressure. We present a first-principles theory of their dynamics, which accounts for the fact that the online population comprises diverse individuals and evolves in time. The resulting equation represents a novel generalization of nonlinear fluid physics and explains the observed behavior across scales. Its shockwave-like solutions explain how, why and when such activity rises from 'out-of-nowhere', and show how it can be delayed, re-shaped and even prevented by adjusting the online collective chemistry. This theory and findings should also be applicable to anti-X activity in next-generation ecosystems featuring blockchain platforms and Metaverses.
Marco Alberto Javarone, Gabriele Di Antonio, Gianni Valerio Vinci, Raffaele Cristodaro · 6 authors
Abstract The behaviour of Bitcoin owners is reflected in the structure and the number of bitcoin transactions encoded in the Blockchain. Likewise, the behaviour of Bitcoin traders is reflected in the formation of bullish and bearish trends in the crypto market. In light of these observations, we wonder if human behaviour underlies some relationship between the Blockchain and the crypto market. To address this question, we map the Blockchain to a spin-lattice problem, whose configurations form ordered and disordered patterns, representing the behaviour of Bitcoin owners. This novel approach allows us to obtain time series suitable to detect a causal relationship between the dynamics of the Blockchain and market trends of the Bitcoin and to find that disordered patterns in the Blockchain precede Bitcoin panic selling. Our results suggest that human behaviour underlying Blockchain evolution and the crypto market brings out a fascinating connection between disorder and panic in Bitcoin dynamics.
Nicolò Vallarano, Matteo Bruno, Emiliano Marchese, Giuseppe Di Trapani · 8 authors
Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years. Rooted into statistical physics, the ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties. From a technical point of view, the ERGMs workflow is defined by two subsequent optimization steps: the first one concerns the maximization of Shannon entropy and leads to identify the functional form of the ensemble probability distribution that is maximally non-committal with respect to the missing information; the second one concerns the maximization of the likelihood function induced by this probability distribution and leads to its numerical determination. This second step translates into the resolution of a system of $O(N)$ non-linear, coupled equations (with $N$ being the total number of nodes of the network under analysis), a problem that is affected by three main issues, i.e. accuracy, speed and scalability. The present paper aims at addressing these problems by comparing the performance of three algorithms (i.e. Newton's method, a quasi-Newton method and a recently-proposed fixed-point recipe) in solving several ERGMs, defined by binary and weighted constraints in both a directed and an undirected fashion. While Newton's method performs best for relatively little networks, the fixed-point recipe is to be preferred when large configurations are considered, as it ensures convergence to the solution within seconds for networks with hundreds of thousands of nodes (e.g. the Internet, Bitcoin). We attach to the paper a Python code implementing the three aforementioned algorithms on all the ERGMs considered in the present work.
In this article, we discuss the random graph, Barabási-Albert (BA) model, and lattice networks from a unified view point, with the parameter $ω$ with values $1,0,-1$ characterizing these networks, respectively. The parameter is related to the preferential attachment of nodes in the networks and has different weights for the incoming and outgoing links. In addition, we discuss the correspondence between quantum statistics and the networks. Positive and negative $ω$ correspond to Bose and Fermi-like statistics, respectively, and we obtain the distribution that connects the two. When $ω$ is positive, it is related to the threshold of Bose-Einstein condensation (BEC). As $ω$ decreases, the area of the BEC phase is narrowed, and disappears in the limit $ω=0$. When $ω$ is negative, nodes have limits in the number of attachments for newly added nodes (outgoing links), which corresponds to Fermi statistics. We also observe the Fermi degeneracy of the network. When $ω=-1$, a standard Fermion-like network is observed. Fermion networks are realized in the cryptocurrency network "Tangle."