Thermodynamic limit

​, which becomes negligible only as $N \to \infty$N→∞. These fluctuations prevent the emergence of sharp phase transitions observed in the infinite-volume limit, instead leading to smoothed or rounded features in observables such as the specific heat, where peaks are broadened rather than infinitely sharp. For instance, in simulations of second-order phase transitions, the specific heat peak rounds out with increasing system size $L$L, with the width scaling as $L^{-1/\nu}$L−1/ν where $\nu$ν is the correlation length exponent, reflecting the dominance of finite-size effects over bulk thermodynamics.^[19][20][21] Boundary effects further undermine the extensivity of thermodynamic properties in finite systems, where surface contributions can dominate over bulk terms, violating the additivity assumed in the thermodynamic limit. In such cases, properties like free energy or entropy do not scale linearly with volume, as interfacial or surface energies introduce non-extensive corrections proportional to the surface area. This is particularly evident in low-dimensional or confined geometries, such as thin films, where the surface-to-volume ratio is high, leading to deviations from bulk behavior; for example, in metallic alloys like Cu-Ni thin films, surface free energy varies with thickness and composition, altering phase stability and thermodynamic potentials. The thermodynamic limit thus breaks down when these boundary-induced non-extensivities persist, as the infinite-volume scaling cannot eliminate surface dominance without additional corrections.^[22][23][24] A prominent example of this inapplicability is Bose-Einstein condensation (BEC) in an ideal Bose gas, which strictly occurs only in the thermodynamic limit of infinite volume $V \to \infty$V→∞ at fixed temperature $T$T below the critical value $T_c$Tc​. In finite volumes, the discrete density of states rounds the transition, shifting the effective critical temperature by an amount $\Delta T_c / T_c \sim (N / \zeta(3/2))^{-1/3}$ΔTc​/Tc​∼(N/ζ(3/2))−1/3, where $N$N is the particle number and $\zeta$ζ is the Riemann zeta function; higher-order corrections further refine this shift, emphasizing that no true condensation occurs for any finite $V$V. Boundary conditions exacerbate this, sensitizing the condensate fraction and transition temperature to system geometry.^[25][26] The recognition of these finite-size effects gained prominence in the 1960s through early Monte Carlo simulations of lattice models near critical points, which revealed systematic deviations from infinite-system predictions and motivated the development of finite-size scaling theory. Pioneering work by researchers like Kurt Binder in the late 1960s and early 1970s demonstrated how simulation data for finite lattices could be extrapolated to the thermodynamic limit by accounting for size-dependent shifts and broadenings, establishing finite-size scaling as a cornerstone for studying critical phenomena in computationally accessible systems.^[27][28]

Beyond Classical Thermodynamics

Historical Development

Origins

Key Contributions and Evolution