Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online
$$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp...
Thanks for your reply. Maybe (probably) I am a little slow, but I dare say I do not see how your comment answers my question.
I know that
A(T, V, N) = −k(B). T. lnZ
and in all textbooks it is shown how this expression coincides with the "classical" one, A = {E} - TS.
My question is, by looking...
I am afraid I still have issues on the matter.
Now I understand that average energy and temperature are related.
So in my canonical System the temperature, and hence the average energy, are constant.
By Definition, the free energy is constant.
The latter equals A = {E} - TS, where {} is used to...
Thank you for your help. So number 2 is the one I misunderstood.
3) is probably bad language from my side. Maye something like, "the probability of the ideal gas to be in a certain microstate with a certain energy E, follows Boltzmann's distribution".
Thanks for your reply but regretfully I do not understand.
The equation was wrong indeed, the correct one is $E = 3NkT/2$.
Having said this, I do not follow your Argument.
You say "
In statistical mechanics, if the system have a well defined temperature, its total energy E must fluctuate", this...
Thanks for your reply.
I am not so sure I understand it though.
While I understand what a mean energy can be, I am not familiar at all with "mean temperature" in the canonical Setting. the temperature as I understand it should be fixed by the bath.
Why is the Expression I quote for energy...
I must be missing some point with regards to the canonical Distribution. Let us imagine I have a closed (to energy and matter) box full of ideal gas at temperature T. The total energy in the box equals hence
E=3N2kT
, where N is the number of molecules, k Boltzmann's connstant.
Next, I allow...
I think the "geometrical interpretation" is the fact that the Ramahujan summation gives the first term in an asymptotic series related to the divergent sum. The matter is explained best here...
Hi All,
reading a paper by Langer (Theory of the Condensation Point, Annals of Physics 41, 108-157, 1967), I came across an analytical continuation technique which I do not understand (would like to upload the paper PDF but I am not so sure this is allowed).
Essentially, he deals with the...
I am not convinced at all that "a thermometer measures only the translational degrees of freedom".
If one goes back to the basics, I think the matter is very clear.
I have a box containing an ideal gas, at a volume, pressure, temperature.
Now I contact it with a thermometer. They will reach...
Hi All,
reading a paper by Langer (Theory of the Condensation Point, Annals of Physics 41, 108-157, 1967), I came across an analytical continuation technique which I do not understand (would like to upload the paper PDF but I am not so sure this is allowed).
Essentially, he deals with the...
Hi all,
I am struggling to grasp the sense of the partition function.
First of all, I had a look at a couple of derivations (which the relevant Wikipedia page follows) in which the concept of heat"energy of a thermal bath" is invoked. Well this is already confusing me: if the thermal bath has an...
DH, thanks a lot for your help.
I checked Wikipedia too and was still not getting it...your explanation involving Binet's formula and the representation of the golden ratio is clear now, bit out of sheer curiosity what is that is being said on Wikipedia about the logarithms, integers...
Hallsoflvy,
regreftully I do not understand what you mean.
Borek's remark was of course correct and I agree with it, as clearly stated in my post. Subsequently I tried to add the (missing) information that would make the problem at least meaningful. i.e. I clarified those "expressions" have...
Borek,
how to disagree with you...I of course forgot to mention the fact those expression have to be perfect squares, i.e. the square of an integer number.
thanks
Hi all, I found the following statement on a magazine page and cannot understand it. It is possibly very distant from the little maths I know, but made me very curious.
It is therein said that if a number $$n$$ is a Fibonacci number, then one of the conditions $$ 5n^2 + 4$$ or $$5n^2-4$$ is...
Hi All,
I am struggling to prove the following identity
$$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$
any hint would the most appreciated. I understand the...
Here you will find a better explanation than I could give on sufficient and necessary conditions for minima http://www.math.utah.edu/~cherk/teach/12calcvar/sec-var.pdf
If you have the book "introduction to Calculus of Variations" by Fox you will find there a thorough discussion of the second...
Given the equation z = f(x,y) and the point (x0,y0,z0) you want to find the direction along which the directional derivative is zero.
The directional derivative as a function of direction (the latter given by a unity vector n, with components n_x and n_y) can be written as
$$\frac{\partial...
Stephen, you are right in pointing out that such notation is confusing. Let me then rephrase my question clearly:
Let us denote a convolution
$$\int_0^{t} A(t-\tau) x(\tau) \mathrm{d}\tau$$
With the notation
$$A \star x$$
I would like to write down the expression for the double convolution...
Well, there is no mention in Wikipedia of Stjeltes convolutions (although they are very strongly related to the measure theory approach described in the Section "Measure"), would it help if I edited my question using Riemann integrals notation?
Let me preface I know very little about Field Theory, so please take my reply with caution.
On the other hand, the step in question seems to me stemming from basic calculus of variations (but care needs to be exterted as there are different definitions of variations around).
If you search...
Stephen,
many thanks for your input.
I am not using an engineering notation, whatever yo mean by that, I am simply writing the convolution down as a Stijeltes convolution (using Stjeltes instead of Riemann integration), as the notation is more compact.
Under certain technical conditions...
Let us write a convolution
$$\int_{0}^{t} A(t-\tau) \mathrm{d}x(\tau)$$ as
$$A \star \mathrm{d}x$$
I would like to write down the expression for the double convolution
$$A \star \mathrm{d}x \star \mathrm{d}x $$
Following the definition I obtain
$$ \int_{0}^{t} \int_{0} ^{t-\tau}...
Hi there,
I got across the integral
$$\int_{\omega} \nabla y(x) \mathrm{d}x$$.
It would be better to perform the integration over the domain $$\Omega$$, the two domains being related by a transformation $$Y:\omega \to \Omega$$.
Using the change of variable rule I wrote
$$\int_{\Omega}...
Thank you ever so much. That is massive progress. The third term, $$-2ln(a)$$ is easily explained, all there is to do now is to understand the origin of the fourth and fifth terms...
They should originate from the integral
$$\int_{-a}^{a} ln (\vert r - \zeta \vert) \mathrm{d}r$$
which I...
Hi All,
I met the following function to evaluate,
$$v(x)=\int_{-\infty}^{-a} G(t) ln(\vert t - x \vert) \mathrm{d}t + \int_{-a}^{a} -N ln(\vert t - x \vert) \mathrm{d}t + \int_{a}^{\infty} G(t) ln(\vert t - x \vert) \mathrm{d}t$$, where G is an unknown even function, N is a constant.
After...
Haeel,
your comment is interesting. It would be certianly the most welcome shouldyou expnad upon it: for example, how would a better notation shed light on the topic of non-commutativity?
thanks a lot
I am reading Cornelius Lanczos' "The Variational principles of Mechanics", and Vujanovic, Atackanovic "Introduction to Modern Variational tecniques in Mechanics and Engineering".
I understand that in some derivations q and its time derivative are treated as independent, that they are to be...
Hi there,
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and...
Hi there,
in a paper the author obtains the integral
$$\int_{a}^{\infty} \frac {g(\lambda(r))}{r}\mathrm{d}r$$
which is claimed to be equivalent to
$$\int_{a/A}^{1} \frac {g(\lambda(r))}{\lambda (\lambda^3-1)}\mathrm{d}\lambda$$
making use of the relationship (previously physically...
Many thanks to all of you for help, very useful and appreciated.
Simon Bridge, yes my solution for the question posed by Arildno was wrong, not a typo but considering g as the constant, again my error.
Thanks again
Arildno,
thanks for your hint. I suggest functions $$f = \sqrt{log x}$$ and $$g = \sqrt{log (x/e^{a^2})}$$.
Simon,
My best attempt so far:
$$\int \frac{1}{(x^2 + (y-s)^2)^2} \mathrm{d}s$$, change variable y-s = z,
$$\int \frac{-1}{(x^2 + z^2)*(x^2+z^2)} \mathrm{d}s =$$
and now I...
I am more and more confused...
So you know Young's modulus! (actually you know the constant of the Neo-Hookean model, is not a Young's modulus strictly speaking, also the formula you reported, Sigma=Youngsmodulus * Epsilon, is not valid at all), you know the stress (you are applying it and it...
No worries for the quick replies, boring day in the office...
Ok maybe I get it know, i was confused by the fact you claimed in the first post wanted to read the value of the Elasticity Modulus in the interphase.
I interpreted this as, I do not know such value in advance.
Well then, how can...
Both F and strain tensor are a function only of gradients of displacement, as is clear from their definition.
Physically this is an essential requirement, the principle of objectivity: moving a body by a rigid motion can not alter its energetic content (which is supposed to be a function of its...
AccessTUD,
finding F is the problem of elasticity itself! Analytically, you have to solve the Navier or Cauchy PDE, not always easy. Numerically, it is the deliverable of a FE computation, for example.
Now back to your point: your example confuses me. In the beam under tension, the strain is...
AccessTUD,
I am unsure I understand the problem.
So you know already the deformation gradient (as a tensor, not only the F(3,3) component I assume) on your domain: your elastic problem is solved already! You wonder then about how to compute the stress tensor: I do not understand why would...
Hi All,
I found it easier to master the functional derivative concept by performing the calculations out, leaving aside for a moment the symbol $$\delta$$ to denote variations, a useful notation that might though hide the mechanics of what is going on.
Then your example becomes transparent...
Let me rephrase the question, to make it clearer.
How to compute the Euler Lagrange equation of the functional
$$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$
where u is a vectorial function, $$\nabla^{(s)}u = \frac{1}{2} (u_{i,j}+u_{j,i})$$ and D is a (symmetric) constant tensor $$D_{ijkl}$$?
Hello there,
I am struggling in proving the following.
The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes
$$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$
among all vectorial functions u satisfying the...
Hi all,
I have a question regarding certain double integrals.
Assume the function $$ l(t) $$ is given as well as the function $$K(t)$$, defined only for positive argument. Also the definition $$n(t) = \int_{-\infty}^{t} K(t-\tau) l(\tau) \mathrm{d}\tau$$ is given. if I wish to compute $$P =...
PS: I think I might be closer now...I just realized what was in front of me clearly, that the fraction is the Green's function for the operator nabla^2, am I right? Thanks a lot to all
Office_shredder,
it looks like I have understood less than what I thought...
Your explanation made perfect sense until I tried a direct computation.
For simplicity in 1D, I fixed y = 1.
Then, if I understood, the indicated function should behave as a delta-distributuion under the integral...