# Questions tagged [functional-derivatives]

Generalization of the notion of derivative to functionals, i.e., to functions that take other functions as an argument. Functional derivatives are particularly useful, for example, in field theory.

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### Using the functional derivative and delta function for proving The Fokker-Planck Equation

I am reading "Lectures on Phase Transitions" by Nigel Goldenfeld, specifically Chapter 8, where the Fokker-Planck equation is derived. I found the following part of the proof, but there are ...
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### Is there a practical distinction between functions of state and functionals in thermodynamics?

In thermodynamics, and more precisely when talking about continuous systems, some sources [1, 2] introduce functionals of state: $$F[s(x), \dots]:=\int_VdV(x)f(s(x),\dots,x)$$ In order to derive ...
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### Vibration of a continuous uniform chain and the normal modes

The question is: A vertically hanged chain with the upper end attached to a fixed point. I try to find the normal modes under the small $\theta$ condition. Consider the mass $\mathrm{d}m$ with ...
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### Do functional derivatives commute in a specific example?

Edit: this question is related to other already asked questions, like Symmetry of second functional derivatives , but a clear and definitive answer has never been given, and I am here giving a ...
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### Is there physical significance to the Gateaux derivative of entropy?

Is there any physical "force" generated by a difference in entropy or information between two regions? There is some confusion between thermodynamic entropy and information entropy, but one ...
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### Functional derivative of a Green function

I'm trying to prove that, given the Hamiltonian $\hat{H} + \int d\mathbf{x} \hat{n}(\mathbf{x})\varphi(\mathbf{x}, t)$, where $\varphi(\mathbf{x}, t)$ is some external field and $\hat{S}$ the ...
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### Evaluation of functional derivative of effective action

I'm trying to understand a calculation in appendix A of this paper https://arxiv.org/abs/2204.04197, however I don't understand how they end up with equation (125) and I think I am going wrong in the ...
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### Calculating Poisson brackets in classical non-relativistic Hamiltonian field theory

Summary of the question: How can I prove the equal-time Poisson bracket relations for the classical Hamiltonian field theory? I.e $$[q(x,t),H(t)]_\mathrm{PB}=\dot{q}(x,t)\tag{1}$$ for a field $q$ and ...
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### Does this particular notation for derivatives imply anything in particular? [duplicate]

In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as: $$\frac{\delta f}{\delta t}$$ Which is a bit strange. Does this ...
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### Fourier transform of a functional derivative

Suppose that $x(q)$ is the Fourier transform of the function $x(r)$, where $r$ is the real-space variable and $q$ is the Fourier-space variable. Then, suppose that $E$ is an energy functional which ...
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### What's the difference between the conjugate momenta in the classical mechanics and in field theory?

In the classical mechanics the conjugate momenta was typically a derivative of the Lagrangian, i.e. $$p_i=\frac{\partial L}{\partial \dot q_i}.\tag{1}$$ However, in the QFT ...
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### Functional derivatives: Fréchet, Gateaux derivatives and Euler-Lagrange operators

What is the relationship between Fréchet derivatives, Gateaux derivatives and the usual Euler-Lagrange operator (ELO)? Is the ELO the Fréchet derivative, the Gateaux derivative or does it depends on ...
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### Is there such a thing as infinitesimal electric field?

I am interested in calculating some response properties, namely, susceptibility and polarizability. In principle, susceptibility should be the functional derivative of the electron density to a ...
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### Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$ with respect to $u$, ($\delta L / \delta u$). The ...