A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (T ${T}_{c}$ ) as well as in the region of the second-order phase transition (T\ensuremath{\simeq} ${T}_{c}$ ), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (T${T}_{c}$) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}${T}_{c}$), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.

Physical review. B, Condensed matterMay 1, 19889 citations

Perspectives from the theory of phase transitions

G. Toulouse

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BSep 1, 1974111 citations

Critical Phenomena and Phase Transitions

John F. Nagle

Springer eBooksJan 1, 197528 citations

The Temporal Scale Invariance Hypothesis as a Bridge between Quantum Mechanics and Gravitation

Alexey Gromov

Abstract A hypothesis is proposed according to which the sole difference between the quantum and classical worlds is the time scale. A quantitative verification of the model has been carried out on systems ranging from atomic to macroscopic scales. The concept of a temporal wake is introduced — a physical field generated by the motion of a particle in its ultrafast proper time. It is shown that this approach not only explains key quantum phenomena (interference, entanglement, tunneling) but also connects them with gravity. Based on the hypothesis, a phenomenological toy model is constructed, demonstrating quantitative agreement with experimental data. Three laws of scale invariance are formulated. A system of coupled field equations is proposed in which the coupling between particles is realized through an effective parameter αeff depending on the time scale, system symmetry, and density. It is found that αeff naturally describes a phase transition between atomic, nuclear, and classical regimes. Testable predictions are formulated, including an experiment on wake shielding (Prediction I) and the search for a phase transition at nuclear density (Prediction L).Abstract A hypothesis is proposed according to which the sole difference between the quantum and classical worlds is the time scale. A quantitative verification of the model has been carried out on systems ranging from atomic to macroscopic scales. The concept of a temporal wake is introduced — a physical field generated by the motion of a particle in its ultrafast proper time. It is shown that this approach not only explains key quantum phenomena (interference, entanglement, tunneling) but also connects them with gravity. Based on the hypothesis, a phenomenological toy model is constructed, demonstrating quantitative agreement with experimental data. Three laws of scale invariance are formulated. A system of coupled field equations is proposed in which the coupling between particles is realized through an effective parameter αeff depending on the time scale, system symmetry, and density. It is found that αeff naturally describes a phase transition between atomic, nuclear, and classical regimes. Testable predictions are formulated, including an experiment on wake shielding (Prediction I) and the search for a phase transition at nuclear density (Prediction L).

Jul 8, 20260 citations

{T}_{c}

) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}

{T}_{c}

), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.