Timeline for Numerically finding a curve that satisfies an underdetermined set of equations
Current License: CC BY-SA 4.0
19 events
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| Sep 5, 2023 at 15:55 | vote | accept | jcp | ||
| Aug 29, 2023 at 14:59 | answer | added | Alex Trounev | timeline score: 3 | |
| Aug 27, 2023 at 14:02 | comment | added | jcp |
But yes, overall solving the constrained minimization of F is equivalent to solving the sets of equations I quoted. However, my question remains, how does one carry that out to reproduce the curves in the paper?
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| Aug 27, 2023 at 14:00 | comment | added | jcp | @AlexTrounev Let us focus on the 2-phase co-existence. You are correct in the equation that you quoted. And this conservation of volume let's us derive the equality of osmotic pressure in each phase equation. However, we additionally need the constraint that the total number of molecules of each type are also conserved. See, "[..] function 𝑓(𝜙,𝜓) contains all of the information needed to characterize the equilibrium state of the system, which minimizes the total free energy of the system, subject to the constraint that the total numbers of molecules of each type are conserved." | |
| Aug 27, 2023 at 8:07 | comment | added | Alex Trounev | Since we have two or more phases we should minimize total free energy. For example, $V_1F(\phi_1,\psi_1)+V_2F(\phi_2,\psi_2)$ for a given volume $V_1+V_2=V$. | |
| Aug 26, 2023 at 21:47 | comment | added | jcp |
i.e directly working with eq. (4) quoted in the main text, equivalent to the F that I have defined?
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| Aug 26, 2023 at 21:24 | comment | added | jcp |
@AlexTrounev I see. Indeed, I think the main objective is to min F subject to constraints that the total number of molecules of each species, is constant. A typical trick is to transform the F (free energy) to the grand potential via Legendre transform (eq. 27 in the SI). Then simple min. of the grand potential yields the conditions on equality of chemical potentials that I quote here. If I understand your suggestion, it is to directly perform the constrained optimization for F instead of using (1),(2) and (3) (or equivalently, the equations I quote?)
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| Aug 26, 2023 at 16:05 | comment | added | Alex Trounev |
I mean that equations (1), (2), (3) are constraints for optimization problem for F. As I understand from discussion about data shown in Figure 2, they simulate some biological noise.
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| Aug 26, 2023 at 12:57 | comment | added | jcp | @AlexTrounev Thank you for your response. I am not quite sure by what you mean. I can get the spinodal which is the det(hessian(F)) already. But the binodal curve would be determined by the equality of chemical potentials? | |
| Aug 26, 2023 at 2:52 | comment | added | Alex Trounev |
We can directly minimize F as well.
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| Aug 25, 2023 at 15:39 | history | edited | jcp | CC BY-SA 4.0 |
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| Aug 25, 2023 at 14:11 | history | edited | jcp | CC BY-SA 4.0 |
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| Aug 25, 2023 at 13:55 | history | edited | jcp | CC BY-SA 4.0 |
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| Aug 25, 2023 at 10:37 | comment | added | jcp | I notice there is a slight mismatch in the notation. in equations (1)-(3) in the paper, the subscript $i$ denotes the phase. | |
| Aug 25, 2023 at 10:34 | comment | added | jcp | Thank you for engaging in discussion. yes, putting $i=1,2$ in those partial derivatives and eliminating the $\eta, \pi $ terms reduces to (39)-(41) from the paper's SI (linked above) or the definitions I have here? | |
| Aug 25, 2023 at 6:43 | comment | added | Alex Trounev | Why not to use equations (1), (2), (3) from the paper cited? | |
| Aug 24, 2023 at 21:41 | history | edited | jcp | CC BY-SA 4.0 |
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| Aug 24, 2023 at 21:11 | history | edited | jcp | CC BY-SA 4.0 |
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| Aug 24, 2023 at 20:43 | history | asked | jcp | CC BY-SA 4.0 |