SciPost Submission Page
Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law
by Loris Di Cairano
Submission summary
| Authors (as registered SciPost users): | Loris Di Cairano |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202512_00024v1 (pdf) |
| Date submitted: | Dec. 10, 2025, 9:12 a.m. |
| Submitted by: | Loris Di Cairano |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We show that thermodynamics can be formulated naturally from the intrinsic geometry of phase space alone---without postulating an ensemble, which instead emerges from the geometric structure itself. Within this formulation, phase transitions are encoded in the geometry of constant-energy manifold: entropy and its derivatives follow from a deterministic equation whose source is built from curvature invariants. As energy increases, geometric transformations in energy-manifold structure drive thermodynamic responses and characterize criticality. We validate this framework through explicit analysis of paradigmatic systems---the 1D XY mean-field model and 2D $\phi^4$ theory---showing that geometric transformations in energy-manifold structure characterize criticality quantitatively. The framework applies universally to long-range interacting systems and in ensemble-inequivalence regimes.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
Weaknesses
I looked through the ref 1 and 2 of Rugh and at first it seemed to me that at least his first paper has the same flaw. He does, however, state beforehand that he makes the choice of a Euclidean phase space to simplify the presentation and later on, that the results only depend upon dH(X)=1, which indeed is coordinate independent.
In view of the above I think that the theoretical geometrical part of the paper falls apart so I can not recommend publication.
Report
Recommendation
Reject
Strengths
- The manuscript provides interesting and new viewpoint on the foundations of statistical physics.
- Mathematical considerations are supported and tested with numerical and analytical analysis of two sample models.
- The paper is nicely organised with all tedious calculations and details unnecessary to follow the main concepts placed in the Supplemental Material.
Weaknesses
- While the topic of the manuscript is interesting for a wide community of physicists, it requires from the reader specific knowledge in the field of symplectic manifolds and differential geometry.
- Many symbols and variables are not properly explained upon their introduction. In most cases the meaning is clear for specialists in the topic but this makes the text harder to follow for other readers.
Report
The advantage of this approach is that it directly links thermodynamic quantities, like temperature or entropy, with geometry of energy shells in $\Lambda$. The Author demonstrates this by deriving the Entropy Flow Equation (EFE, Eq. 7) and checks its validity against numerical calculations for two sample models. Finally, for these two sample models, the Author analyses the phase transition at the critical point from the point of view of geometry of energy shells.
I have found this paper very interesting and discussion-inspiring. In my opinion, it clearly fulfils the conditions for publishing in SciPost Physics. However, there are some things that could be improved.
The main problem of the manuscript is that, while it discusses a topic of interest for a wide community of physicists, it requires from the reader very specific knowledge of several mathematical concepts. This makes it very difficult to follow for somebody working in the field of statistical physics but not skilled in symplectic manifolds and differential geometry (including myself). Of course, this issue is impossible to fully resolve as mathematical formalism is absolutely necessary here. Nevertheless, in my opinion it is important to make the manuscript as simple to follow for non-specialists as possible.
The Author partially addressed this problem by organising the text such that parts with tedious mathematical calculations are grouped in the Supplemental Material, and can be easily skipped without loosing any ideas presented in the manuscript. However, there still are a few things that can be done in this matter. Below, I have listed several minor changes that would make the reading easier; they mostly consists of proper defining of used quantities (meaning of which is probably clear for experts in the field) and clarifying some confusing wording. The Author is kindly asked to go through the whole text and improve it having proper definitions and clarity of text in mind.
Another issue is with the geometric reparametrisation of phase space. Different parametrisations clearly lead to different forms of the metric tensor $\eta$. The Author claims that all possible metric tensors give the same thermodynamics and call this by "thermodynamic covariance". However, I have not found a proof of this anywhere in the manuscript. This needs to be clarified.
I am also curious what happens when $\nabla^\eta H=0$ at some point of phase space? Is this case impossible from physical point of view? Or is there a way to construct in this case metric tensor $g$, anyway?
Finally, I have a question related to phase transitions. In the manuscript the Author shows that, for the two sample models they consider, the critical point coincides with a reversal of curvature in the energy manifolds. Is this a general property of critical points? Or are there any other geometrical mechanisms leading to criticality? A comment on this matter would be very valuable.
To conclude, after resolving the described issues, I recommend publication of this interesting manuscript in SciPost Physics journal.
Requested changes
Main issues (as described in more detail in the report):
A1. Please clarify the reason why Euclidean metric tensors $\eta$ for different representations of phase space belong to the same universality class and, therefore, they give the same microcanonical measure.
A2. Please add a comment about the case of $\nabla^\eta H=0$.
A3. Please add a comment on geometrical mechanisms leading to criticality.
Minor issues (mostly adding definitions of used quantities and clarifying the description; I am aware that some of the changes can be seen as unnecessary, but they really help non-experts to go through the manuscript):
B1. Please add equation numbers to all formulae in the manuscript and SM. This will simplify future cross referencing.
B2. The order of sections in SM does not agree with the order in which they are referenced in the main text.
B3. Page 1, left column, paragraph 1: I suggest adding a reference the first time symplectic structure is mentioned.
B4. Page 1, left column, paragraph 1 and 2: Please state that the considered evolution of entropy is not a time evolution.
B5. Page 1, left column: Please consider defining the meaning of symbols $\Lambda$, $H$, $\omega$, and $\eta$.
B6. Page 1, right column, paragraph 2: The EFE is not solved exactly but numerically.
B7. Page 1, right column, first equation: Please consider adding explanation that the symbol "$\iota$" denotes the interior product.
B8. Page 2, left column, paragraph 1: Please add that Einstein summation convention is used.
B9. Page 2, left column, just above second equation: The term "is converted into" is unclear; I propose "takes form of" instead.
B10. Page 2, left column, before fourth equation: The statement "generates the motion of points on an energy hypersurface to others" is not clear; consider changing "others" to "other hypersurfaces".
B11. Page 2, right column, below second equation: The phrase "passes from right to left" is unclear; is it right-hand-side and left-hand-side of some equation?
B12. Page 3, left column, Equation (1): It is not clear what index $B$ in $\Omega_B$ and $S_B$ refers to.
B13. Page 9, Sec. B: In first there equations, the gradient $\nabla$ has to metric tensor $\eta$ in index. It only appears in fourth equation on page 10. Is this correct?
B14. Page 10, Sec. C: In Eq. (S15) and one equation before there is no index $\eta$ in $d\sigma_E$. Is this correct?
B15. Page 13, Sec. F: The section needs a short introduction to explain what is actually calculated and the meaning of symbols.
B16. Page 13, Sec. F: The symbol "$I_k$" should be defined as a modified Bessel function of the first kind after first equation.
Misprints:
C1. Page 1, second column, paragraph 1: "stimulating an debate" should be replaced with "stimulating a debate".
C2. Page 2, first column, bottom: "$y^\alpha$ coordinates on $\Lambda$" should be replaced with "$y^\alpha$ coordinates on $\Sigma_E$".
C3. Page 4, second column: All references to Figure S2 should be replaced with references to Figure 1.
Recommendation
Ask for minor revision