# A Problem in Thermodynamics and Statistical Mechanics: Analytical and Numerical Study of an Einstein Solid

Every physics major at some point in their undergraduate career takes a course in thermodynamics and statistical mechanics. One of my problem sets included a problem that considers an Einstein solid with 50 oscillators and 100 units of energy and then increases the number of oscillators to 5000. I will be presenting my solution to the numerical side of the problem. An Einstein solid can be regarded as

“… a collection of microscopic systems which can store any number of energy ‘units’ of equal size which occur for any quantum-mechanical harmonic oscillator whose potential energy function has the form $\displaystyle \frac{1}{2}k_{s}x^{2}$…The model of a solid as a collection of identical oscillators with quantized energy units…”

described (defined) by Schroeder in his text Introduction to Thermal Physics. Figure 1 represents the Einstein solid as a whole (in a lattice) and Figure 2 depicts the quantum-mechanical harmonic oscillator interpretation of an Einstein solid.

The problem statement is:

“Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy…Make a graph of entropy vs. energy, and a graph of the heat capacity vs. temperature. Then change the number of oscillators to 5000, and again make a graph of the heat capacity and temperature and entropy and energy, and discuss the predictions and compare it to the predictions to the data for lead, aluminum, and diamond. Estimate the numerical value of $\displaystyle \epsilon$ for each of those solids.”

This problem can be found in the aforementioned text.

Figure 1. Einstein Solid (Lattice); Image Credit/Obtained from https://mappingignorance.org/2015/12/17/einstein-and-quantum-solids/

Figure 2. Quantum-Mechanical Harmonic Oscillator interpretation of an Einstein solid as a collection of these oscillators. Image Credit: http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/einsol.html

Part I: Let $q = 100$ units, and let $N = 50$. The corresponding data table for this Einstein solid follows. The following set of equations were used to determine the multiplicity and entropy.

$\displaystyle \Omega(N,q) = {q+N-1 \choose q} = \frac{(q+N-1)!}{q! (N-1)!}, (1)$

and

$\displaystyle S = Nk \ln{\Omega}, (2)$

where $\Omega$ is the multiplicity. The remaining quantities of temperature were obtained using a simplified form of the central difference equations for the first order derivative. The respective definitions of temperature and heat capacity are

$\displaystyle T = \frac{\partial U}{\partial S}, (3)$

and

$\displaystyle C_{V} = \frac{\partial U}{\partial T}, (4)$

where $U$ represents the internal energy of the Einstein solid, and $S$ is the entropy. The generalized from of the first order central difference approximation has the form

$\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j+1}-y_{j-1}}{2h} + \mathcal{O}(h^{2}), (5)$

where $\mathcal{O}(h^{2})$ represents the higher order terms, in this case, the quadratic, cubic, quartic, and so on, and $h$ is the step size for each iteration. For the final iteration (when $q = 100$ units), instead of using the central difference approximation, a backward difference approximation was employed since there does not exist data for $q = 101$ units of energy. The backward difference approximation has the form

$\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j}-y_{j-1}}{h}+\mathcal{O}(h). (6)$

Table I (Dimensionless Parameters):

 Energy q Ω S/k kT/ε C/Nk 0 1 0 0 N/A 1 50 3.912023005 0.27969284 0.121826198 2 1275 7.150701458 0.328336604 0.453606383 3 22100 10.00333289 0.367875021 0.536183525 4 292825 12.58733044 0.402937926 0.593741905 5 3162510 14.96687657 0.43524436 0.637773801 6 28989675 17.18245029 0.465656087 0.673043377 7 231917400 19.26189183 0.494675894 0.702124659 8 1652411475 21.22550156 0.522626028 0.72660015 9 10648873950 23.08871999 0.549726805 0.747522024 10 62828356305 24.86367234 0.576136157 0.765628174 11 3.427E+11 26.56012163 0.601971486 0.781456694 12 1.74206E+12 28.18608885 0.627322615 0.795411957 13 8.30828E+12 29.74827387 0.652259893 0.807805226 14 3.73873E+13 31.25235127 0.676839501 0.818880855 15 1.59519E+14 32.70318415 0.701107048 0.828833859 16 6.48046E+14 34.1049827 0.725100078 0.837822083 17 2.51594E+15 35.4614241 0.748849881 0.845974847 18 9.3649E+15 36.77574496 0.772382808 0.853399232 19 3.35165E+16 38.05081369 0.795721261 0.860184741 20 1.15632E+17 39.28918792 0.818884446 0.866406816 21 3.8544E+17 40.49316072 0.84188895 0.872129523 22 1.24392E+18 41.66479814 0.864749193 0.877407641 23 3.89401E+18 42.80597005 0.887477794 0.882288296 24 1.18443E+19 43.91837566 0.910085848 0.88681226 25 3.5059E+19 45.00356493 0.932583169 0.891014994 26 1.01132E+20 46.0629565 0.954978471 0.89492748 27 2.84667E+20 47.09785298 0.977279528 0.898576916 28 7.82835E+20 48.10945389 0.999493303 0.901987268 29 2.10556E+21 49.09886689 1.021626052 0.905179739 30 5.54463E+21 50.06711736 1.043683421 0.908173155 31 1.43087E+22 51.01515679 1.065670516 0.910984284 32 3.6219E+22 51.94387004 1.087591972 0.913628113 33 8.99987E+22 52.85408172 1.109452006 0.916118071 34 2.19703E+23 53.74656181 1.131254467 0.918466228 35 5.27286E+23 54.62203054 1.153002873 0.920683463 36 1.24498E+24 55.48116286 1.174700452 0.9227796 37 2.89374E+24 56.32459225 1.196350168 0.924763536 38 6.62514E+24 57.1529142 1.217954752 0.926643346 39 1.4949E+25 57.96668937 1.239516722 0.928426373 40 3.32616E+25 58.76644629 1.261038406 0.930119309 41 7.30133E+25 59.55268389 1.282521958 0.931728264 42 1.58195E+26 60.32587378 1.303969378 0.933258829 43 3.38465E+26 61.08646224 1.325382522 0.934716125 44 7.15391E+26 61.8348721 1.346763119 0.936104855 45 1.49437E+27 62.57150439 1.368112778 0.937429341 46 3.08621E+27 63.29673989 1.389433002 0.938693566 47 6.30374E+27 64.01094048 1.410725193 0.9399012 48 1.27388E+28 64.71445045 1.431990666 0.941055635 49 2.54776E+28 65.40759763 1.45323065 0.942160007 50 5.04457E+28 66.09069447 1.4744463 0.943217222 51 9.89131E+28 66.76403902 1.495638697 0.944229971 52 1.9212E+29 67.42791582 1.516808861 0.945200757 53 3.6974E+29 68.08259672 1.537957749 0.946131903 54 7.05244E+29 68.72834166 1.559086264 0.947025573 55 1.33355E+30 69.36539938 1.580195257 0.947883783 56 2.50041E+30 69.99400804 1.60128553 0.948708411 57 4.64989E+30 70.61439586 1.622357843 0.949501213 58 8.57824E+30 71.22678169 1.643412912 0.95026383 59 1.57025E+31 71.83137547 1.664451416 0.950997796 60 2.85263E+31 72.42837879 1.685473998 0.951704548 61 5.14408E+31 73.01798529 1.706481267 0.952385432 62 9.20957E+31 73.60038111 1.7274738 0.953041713 63 1.63726E+32 74.17574525 1.748452147 0.953674575 64 2.89078E+32 74.74424999 1.769416829 0.954285135 65 5.06999E+32 75.30606117 1.79036834 0.95487444 66 8.83407E+32 75.86133855 1.811307153 0.955443478 67 1.52948E+33 76.41023613 1.832233716 0.955993177 68 2.63161E+33 76.95290236 1.853148456 0.956524415 69 4.50043E+33 77.48948048 1.874051781 0.957038017 70 7.65073E+33 78.02010873 1.894944079 0.957534764 71 1.29308E+34 78.54492059 1.915825721 0.958015394 72 2.17309E+34 79.06404502 1.936697061 0.958480604 73 3.63175E+34 79.57760662 1.957558438 0.958931052 74 6.03655E+34 80.08572588 1.978410175 0.959367363 75 9.98043E+34 80.58851934 1.999252581 0.959790129 76 1.64152E+35 81.08609973 2.020085953 0.960199908 77 2.68612E+35 81.57857622 2.040910573 0.960597234 78 4.37356E+35 82.06605448 2.061726714 0.960982609 79 7.08627E+35 82.54863689 2.082534635 0.961356514 80 1.14266E+36 83.02642266 2.103334587 0.961719401 81 1.8339E+36 83.49950796 2.124126809 0.962071705 82 2.92977E+36 83.96798603 2.14491153 0.962413835 83 4.65939E+36 84.43194735 2.165688971 0.962746183 84 7.37736E+36 84.89147968 2.186459344 0.963069122 85 1.16302E+37 85.34666822 2.207222854 0.963383006 86 1.82567E+37 85.7975957 2.227979695 0.963688173 87 2.85392E+37 86.24434247 2.248730056 0.963984945 88 4.44304E+37 86.68698658 2.269474119 0.964273631 89 6.8892E+37 87.12560389 2.290212057 0.964554522 90 1.064E+38 87.56026816 2.31094404 0.9648279 91 1.63692E+38 87.99105107 2.331670228 0.965094032 92 2.50876E+38 88.41802239 2.352390779 0.965353173 93 3.83058E+38 88.84124995 2.373105841 0.965605568 94 5.82737E+38 89.2607998 2.39381556 0.965851451 95 8.83307E+38 89.67673621 2.414520077 0.966091045 96 1.33416E+39 90.08912176 2.435219526 0.966324564 97 2.00812E+39 90.4980174 2.455914037 0.966552213 98 3.01218E+39 90.90348251 2.476603736 0.966774189 99 4.50306E+39 91.30557493 2.497288745 1.287457337 100 6.70955E+39 91.70435105 2.507672727 1.926043463

Graphing the entropy vs. energy, and the heat capacity vs. temperature gives the following:

Graphs I & II

Part II: Let $q = 100$ units and let $N = 5000$. Using this in the calculation yields the following table for this Einstein solid. This “dilutes” the system and lowers the temperature:

Table II (Dimensionless  Parameters):

 Energy q Ω S/k kT/ε C/Nk 0 1 0 0 N/A 1 5000 8.517193 0.122388 0.003049 2 12502500 16.34144 0.131206 0.026553 3 2.08E+10 23.76042 0.137453 0.035575 4 2.61E+13 30.89192 0.14245 0.043342 5 2.61E+16 37.80047 0.146681 0.050387 6 2.18E+19 44.52691 0.150388 0.056922 7 1.56E+22 51.09939 0.153709 0.063064 8 9.74E+24 57.53854 0.156731 0.068885 9 5.42E+27 63.86011 0.159515 0.074436 10 2.72E+30 70.07651 0.162105 0.079755 11 1.24E+33 76.19781 0.164531 0.08487 12 5.16E+35 82.23229 0.166818 0.089804 13 1.99E+38 88.18693 0.168985 0.094575 14 7.13E+40 94.06767 0.171047 0.099198 15 2.38E+43 99.87961 0.173017 0.103687 16 7.47E+45 105.6272 0.174905 0.108052 17 2.2E+48 111.3144 0.176719 0.112303 18 6.14E+50 116.9446 0.178467 0.116448 19 1.62E+53 122.5209 0.180154 0.120494 20 4.07E+55 128.0462 0.181787 0.124447 21 9.73E+57 133.5229 0.183368 0.128314 22 2.22E+60 138.9532 0.184904 0.132098 23 4.85E+62 144.3393 0.186396 0.135806 24 1.02E+65 149.683 0.187849 0.13944 25 2.04E+67 154.9861 0.189265 0.143004 26 3.94E+69 160.2502 0.190646 0.146503 27 7.34E+71 165.4768 0.191995 0.149939 28 1.32E+74 170.6671 0.193314 0.153316 29 2.28E+76 175.8226 0.194604 0.156635 30 3.83E+78 180.9444 0.195868 0.159899 31 6.21E+80 186.0336 0.197106 0.16311 32 9.77E+82 191.0912 0.19832 0.166272 33 1.49E+85 196.1183 0.199512 0.169384 34 2.21E+87 201.1157 0.200682 0.172451 35 3.17E+89 206.0843 0.201831 0.175472 36 4.44E+91 211.025 0.202961 0.17845 37 6.04E+93 215.9384 0.204073 0.181387 38 8E+95 220.8254 0.205166 0.184283 39 1.03E+98 225.6866 0.206243 0.18714 40 1.3E+100 230.5227 0.207304 0.18996 41 1.6E+102 235.3343 0.208349 0.192743 42 1.9E+104 240.122 0.209379 0.195491 43 2.3E+106 244.8863 0.210395 0.198205 44 2.6E+108 249.6279 0.211397 0.200885 45 2.9E+110 254.3472 0.212386 0.203533 46 3.2E+112 259.0447 0.213363 0.20615 47 3.4E+114 263.7209 0.214327 0.208736 48 3.6E+116 268.3762 0.215279 0.211293 49 3.7E+118 273.0112 0.21622 0.213821 50 3.7E+120 277.6261 0.21715 0.21632 51 3.7E+122 282.2214 0.218069 0.218793 52 3.6E+124 286.7975 0.218978 0.221238 53 3.4E+126 291.3548 0.219877 0.223658 54 3.2E+128 295.8935 0.220766 0.226052 55 2.9E+130 300.4141 0.221646 0.228422 56 2.7E+132 304.9169 0.222517 0.230768 57 2.4E+134 309.4022 0.22338 0.23309 58 2.1E+136 313.8703 0.224233 0.235388 59 1.8E+138 318.3214 0.225079 0.237665 60 1.5E+140 322.756 0.225917 0.239919 61 1.2E+142 327.1743 0.226746 0.242152 62 1E+144 331.5765 0.227568 0.244363 63 8.1E+145 335.9628 0.228383 0.246555 64 6.4E+147 340.3337 0.229191 0.248725 65 5E+149 344.6892 0.229991 0.250876 66 3.8E+151 349.0296 0.230785 0.253008 67 2.9E+153 353.3553 0.231572 0.255121 68 2.2E+155 357.6663 0.232353 0.257215 69 1.6E+157 361.9629 0.233127 0.259291 70 1.1E+159 366.2453 0.233896 0.261349 71 8.2E+160 370.5137 0.234658 0.263389 72 5.8E+162 374.7683 0.235414 0.265413 73 4E+164 379.0093 0.236165 0.267419 74 2.7E+166 383.237 0.23691 0.269409 75 1.9E+168 387.4514 0.23765 0.271382 76 1.2E+170 391.6527 0.238384 0.273339 77 8.2E+171 395.8412 0.239113 0.275281 78 5.3E+173 400.0169 0.239837 0.277207 79 3.4E+175 404.1802 0.240556 0.279118 80 2.2E+177 408.331 0.24127 0.281015 81 1.4E+179 412.4696 0.24198 0.282896 82 8.4E+180 416.5962 0.242684 0.284763 83 5.2E+182 420.7108 0.243384 0.286616 84 3.1E+184 424.8136 0.24408 0.288455 85 1.9E+186 428.9048 0.244771 0.29028 86 1.1E+188 432.9845 0.245458 0.292092 87 6.5E+189 437.0529 0.24614 0.29389 88 3.7E+191 441.11 0.246819 0.295676 89 2.1E+193 445.156 0.247493 0.297448 90 1.2E+195 449.191 0.248164 0.299208 91 6.7E+196 453.2152 0.24883 0.300955 92 3.7E+198 457.2286 0.249493 0.30269 93 2E+200 461.2315 0.250152 0.304413 94 1.1E+202 465.2238 0.250807 0.306124 95 5.9E+203 469.2057 0.251458 0.307823 96 3.2E+205 473.1774 0.252106 0.30951 97 1.7E+207 477.1389 0.252751 0.311187 98 8.6E+208 481.0903 0.253392 0.312851 99 4.4E+210 485.0318 0.254029 0.418439 100 2.3E+212 488.9634 0.254345 0.628243

Thus the graphs of the entropy vs. energy and heat capacity vs. temperature follow:

Figure 2. Graphs III and IV.

Figure 3. (Figure 1.14 of Schroeder’s Thermal Physics) Heat Capacity curves for Lead (Pb), Aluminum (Al), and Diamond, respectively as a function of temperature in Kelvin.

Graph II shows the prediction for heat capacity as a function of temperature of an Einstein solid for which there are 100 units of energy and 50 oscillators. The data exhibits a trend that appears to reach an asymptote quickly, then when the temperature reaches T ≈ 2.5, there is a sudden increase in the value of the heat capacity. The approach to determining the final data points was switched from a central difference approximation to a backward difference approximation of the last two entries corresponding to energies q = 99 and q = 100 units. If we ignore the last two, the curve approaches an asymptote at CV = 1. However, the graphs produced are of the dimensionless quantities involved. The overall curve appears to be logarithmic and resembles the heat capacity curve for lead. The initial increase is almost immediate and its slope appears to be slightly less than lead but greater than aluminum.

Graphs III and IV show the prediction for heat capacity in terms of temperature of an Einstein solid for which the energy is the same, but the number of oscillators is now 5000. The temperature has been reduced and the heat capacity vs. temperature yields a graph that shows a trendline that appears linear. Comparing to Figure 3(Fig. 1.14 in the text), this graph resembles the heat capacity curve for diamond. In Figure 3, the diamond curve is linear throughout. The only discrepancies among Graph IV and Figure 3 are the final two data points in Graph IV. Again, a backward difference approximation was used to determine the final data points for this Einstein solid as well.  The value for the constant ε was determined by finding the quotient of the entropy and temperature columns and taking the average value of ε for each energy.

This was the numerical analysis of an Einstein solid’s temperature, energy, entropy, and heat capacity. In the next post, I shall discuss the analytical version of this analysis.

# Coordinates and Transformations of Coordinates

SOURCES FOR CONTENT:  Neuenschwander, D.E., Tensor Calculus for Physics, 2015. Johns Hopkins University Press.

In this post, I continue the introduction of tensor calculus by discussing coordinates and coordinate transformations as applied to relativity theory. (A side note: I have acquired Misner, Thorne, and Wheeler’s Gravitation and will be using it sparingly given its reputation and weight of the material.)

# Derivation of the Finite-Difference Equations

In my final semester, my course load included a graduate course that had two modules: astronomical instrumentation and numerical modeling. The latter focused on developing the equations of motion of geophysical fluid dynamics (See Research in Magnetohydrodynamics). Such equations are then converted into an algorithm based on a specific type of numerical method of solving the exact differential equation.

The purpose of this post is to derive the finite-difference equations. Specifically, I will be deriving the forward, backward, centered first order equations. We start with the Taylor expansion about the points $x_{0}= \pm h$:

$\displaystyle f(x+h)=\sum_{n=1}^{\infty}\frac{h^{n}}{n!}\frac{d^{n}f}{dx^{n}}, (1)$

and

$\displaystyle f(x-h)=\sum_{n=1}^{\infty}(-1)^{n}\frac{h^{n}}{n!}\frac{d^{n}f}{dx^{n}}. (2)$

Let $f(x_{j})=f_{j}, f(x_{j}+h)=f_{j+1}, f(x_{j}-h)=f_{j-1}$. Therefore, if we consider the following differences…

$\displaystyle f_{j+1}-f_{j}=f^{\prime}_{j}+f^{\prime \prime}_{j}\frac{h^{2}}{2!}+...+f^{n}_{j}\frac{h^{n}}{n!}, (3)$

and

$\displaystyle f_{j}-f_{j-1}=hf^{\prime}_{j}-\frac{h^{2}}{2!}f^{\prime \prime}_{j}+...\mp \frac{h^{n}}{n!}f^{n}_{j}, (4)$

and

$\displaystyle f_{j+1}-f_{j-1}=2hf^{\prime}_{j}+\frac{2h^{3}}{3!}f^{\prime\prime\prime}_{j}+..\mp \frac{h^{n}}{n!}f^{n}_{j}, (5)$

and if we keep only linear terms, we get

$\displaystyle f^{\prime}_{j}=\frac{f_{j+1}-f_{j}}{h}+\mathcal{O}(h), (6)$

$\displaystyle f^{\prime}_{j}=\frac{f_{j}-f_{j-1}}{h}+\mathcal{O}(h), (7)$

and

$\displaystyle f^{\prime}_{j}=\frac{f_{j+1}-f_{j-1}}{2h}+\mathcal{O}(h)$

where the first is the forward difference, the second is the backward difference, the last is the centered difference, and $\mathcal{O}(h)$ represents the quadratic, cubic, quartic,quintic,etc. terms. One can use similar logic to derive the second-order finite-difference equations.

# Helicity and Magnetic Helicity

SOURCE FOR CONTENT: Davidson, P.A., 2001, An Introduction to Magnetohydrodynamics. 3.

The purpose of this post is to introduce the concepts of helicity as an integral invariant and its magnetic analog. More specifically, I will be showing that

$\displaystyle h = \int_{V_{\omega}}\textbf{u}\cdot \omega d^{3}r (1)$

is invariant and thus correlates to the conservation of vortex line topology using the approach given in the source material above.  I will then discuss magnetic helicity and how it relates to MHD.

Consider the total derivative of $\textbf{u}\cdot \omega$:

$\displaystyle \frac{D}{Dt}(\textbf{u}\cdot \omega) = \frac{D\textbf{u}}{Dt}\cdot \omega + \frac{D\omega}{Dt}\cdot \textbf{u}, (2)$

where I have used the product rule for derivatives. Also I denote a total derivative as $\frac{D}{Dt} = \frac{\partial}{\partial t}+(\nabla \cdot \textbf{v})$ so as not to confuse it with the notation for an ordinary derivative. Recall the Navier-Stokes’ equation and the vorticity equation

$\displaystyle \frac{\partial \textbf{u}}{\partial t}+(\nabla \cdot \textbf{v})\textbf{u}=\textbf{F}-\frac{1}{\rho}\nabla P + \nu \nabla^{2}\textbf{u}, (3)$

and

$\displaystyle \frac{\partial \omega}{\partial t}+(\nabla \cdot \textbf{v})\omega =(\nabla \cdot \omega)\textbf{u} +\eta \nabla^{2}\omega (4).$

Let $\textbf{F}=0$ and let $\eta \nabla^{2} \omega \equiv 0$ and $\nu \nabla^{2} \textbf{u} \equiv 0$. Then,

$\displaystyle \frac{D\textbf{u}}{Dt}=-\nabla \frac{P}{\rho}, (5)$

and

$\displaystyle \frac{D\omega}{Dt}=(\nabla \cdot \omega)\textbf{u}. (6)$

If we substitute Eqs. (5) and (6) into Eq.(2), we get

$\displaystyle \frac{D}{Dt}(\textbf{u}\cdot \omega)=-\nabla \frac{P}{\rho}\cdot \omega + (\nabla \cdot \omega)\textbf{u}\cdot \textbf{u}. (7)$

Evaluating each scalar product gives

$\displaystyle -2\nabla \frac{P}{\rho}\omega + \omega u^{2}\nabla = 0. (8)$

If we divide by 2 and note the definition of the scalar product, we may rewrite this as follows

$\displaystyle \nabla \cdot \bigg\{-\frac{P}{\rho}\omega +\frac{u^{2}}{2}\omega\bigg\}=0. (9)$

With this in mind, consider now the total derivative of $(\textbf{u}\cdot \omega)$ multiplied by a volume element $\delta V$ given by

$\displaystyle \frac{D}{Dt}(\textbf{u}\cdot \omega)\delta V = \nabla \cdot \bigg\{-\frac{P}{\rho}\omega +\frac{u^{2}}{2}\omega\bigg\}\delta V. (10)$

When the above equation is integrated over a volume $V_{\omega}$, the total derivative becomes an ordinary time derivative. Thus, Eq.(10) becomes

$\displaystyle \int_{V_{\omega}}\frac{d}{dt}(\textbf{u}\cdot \omega)dV=\iint_{S_{\omega}}\bigg\{\nabla \cdot [-\frac{P}{\rho}\omega + \frac{u^{2}}{2}\omega ]\bigg\}\cdot d\textbf{S}=0. (11)$

Hence, Eq.(11) shows that helicity is an integral invariant. In this analysis, we have assumed that the fluid is inviscid. If one considers two vortex lines (as Davidson does as a mathematical example), one finds that there exists two contributions to the overall helicity, corresponding to an individual vortex line. If one considers the vorticity multiplied by a volume element, one finds that for each line, their value helicity is the same. Thus, because they are the same, we can say that the overall helicity remains invariant and hence are linked. If they weren’t linked, then the helicity would be 0. In regards to magnetic helicity we define this as

$\displaystyle h_{m}=\int_{V_{B}}\textbf{A}\cdot \textbf{B}dV, (12)$

where $\textbf{A}$ is the magnetic vector potential which is described by the following relations

$\displaystyle \nabla \times \textbf{A}=\textbf{B}, (13.1)$

and

$\displaystyle \nabla \cdot \textbf{A}=0. (13.2)$

Again, we see that magnetic helicity is also conserved. However, this conservation arises by means of Alfven’s theorem of flux freezing in which the magnetic field lines manage to preserve their topology.

# “Proof” of Alfven’s Theorem of Flux Freezing

SOURCE FOR CONTENT: Choudhuri, A.R., 2010. Astrophysics for Physicists. Ch. 8.

In the previous post we saw the consequences of different regimes of the magnetic Reynolds’ number under which either diffusion or advection of the magnetic field dominates. In this post, I shall be doing a “proof” of Alven’s Theorem of Flux Freezing. (I hesitate to call it a proof since it lacks the mathematical rigor that one associates with a proof.) Also note in this post, I will be working with the assumption of a high magnetic Reynolds number.

Alfven’s Theorem of Flux Freezing: Suppose we have a surface $S$ located within a plasma at some initial time $t_{0}$. From the theorem it is known that the flux of the associated magnetic field is linked with surface $S$ by

$\displaystyle \int_{S}\textbf{B}\cdot d\textbf{S}. (1)$

At some later time $t^{\prime}$, the elements of plasma contained within $S$ at $t_{0}$ move to some other point and will constitute some different surface $M$. The magnetic flux, linked to $M$ at $t^{\prime}$ by

$\displaystyle \int_{M}\textbf{B}\cdot d\textbf{M}, (2)$

from which we may mathematically state the theorem as

$\displaystyle \int_{S}\textbf{B}\cdot d\textbf{S}=\int_{M}\textbf{B}\cdot d\textbf{M}. (3)$

If we know that the magnetic field evolves in time in accordance to the induction equation we may express Eq.(3) as

$\displaystyle \frac{d}{dt}\int_{S}\textbf{B}\cdot d\textbf{S}=0. (4)$

To confirm that this is true, we note the two ways magnetic flux may change as being due to either (1) some intrinsic variability of the magnetic field strength or (2) movement of the surface. Therefore, either way it follows that

$\displaystyle \frac{d}{dt}\int_{S}\textbf{B}\cdot d\textbf{S}=\int_{S}\frac{\partial \textbf{B}}{\partial t}\cdot d\textbf{S}+\int_{S}\textbf{B}\cdot \frac{d}{dt}(d\textbf{S}). (5)$

Now, consider again the two surfaces. Let us suppose now that $M$ is displaced some distance relative to $S$. Further, let us also suppose that this displacement occurs during a time interval $t^{\prime}=t_{0}+\delta t.$ Additionally, if we imagine a cylinder formed by projecting a circular cross-section from one surface to the other, we may consider its length to be $\delta l$ with area given by the cross product: $-\delta t \textbf{v}\times \delta\textbf{l}$. Moreover, since we know that the area of integration is a closed region we see that the integral vanishes (goes to 0). Thus, we may write the difference

$\displaystyle d\textbf{M}-d\textbf{S}-\delta \oint \textbf{v}\times \delta \textbf{l}=0 (6).$

Recall the definition for a derivative, we may apply it to the second term on the right hand side of Eq.(5) to get

$\displaystyle \frac{d}{dt}(d\textbf{S})=\lim_{\delta t\rightarrow 0}\frac{d\textbf{M}-d\textbf{S}}{\delta t}=\oint \textbf{v}\times \delta \textbf{l}. (7)$

Thus the term becomes

$\displaystyle \int_{S}\textbf{B}\cdot \frac{d}{dt}(d\textbf{S})=\int \oint \textbf{B}\cdot (\textbf{v}\times \delta\textbf{l})=\int \oint (\textbf{B}\times \textbf{v})\cdot \delta l. (8)$

Since the integrals that exist interior to the boundary of the surface (call it path $C$) vanish and recall Stokes’ theorem

$\displaystyle \oint_{\partial \Omega}\textbf{F}\cdot d\textbf{r}=\int\int_{\Omega} (\nabla \times \textbf{F})\cdot d\Omega,$

and applying it to Eq.(8) we arrive at

$\displaystyle \frac{d}{dt}\int_{S}\textbf{B}\cdot d\textbf{S}=\oint_{\partial S}(\textbf{B}\times\textbf{v}) \cdot \delta \textbf{l}=\int\int_{S}\bigg\{\nabla \times (\textbf{B}\times\textbf{v})\bigg\}\cdot d\textbf{S}. (9)$

Recall that we are dealing with high magnetic Reynolds number, if we use the corresponding form of the induction equation in Eq.(9) we arrive at

$\displaystyle \frac{d}{dt}\int_{S}\textbf{B}\cdot d\textbf{S}=\int\int_{S}d\textbf{S}\cdot \bigg\{ \frac{\partial \textbf{B}}{\partial t}-\nabla\times (\textbf{v}\times \textbf{B})\bigg\}= 0. (10)$

Thus, this completes the “proof”.

# Deriving the Bessel Function of the First Kind for Zeroth Order

NOTE: I verified the solution using the following text: Boyce, W. and DiPrima, R. Elementary Differential Equations.

In this post, I shall be deriving the Bessel function of the first kind for the zeroth order Bessel differential equation. Bessel’s equation is encountered when solving differential equations in cylindrical coordinates and is of the form

$\displaystyle x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+(x^{2}-\nu^{2})y(x)=0, (1)$

where $\nu = 0$ describes the order zero of Bessel’s equation. I shall be making use of the assumption

$\displaystyle y(x)=\sum_{j=0}^{\infty}a_{j}x^{j+r}, (2)$

where upon taking the first and second order derivatives gives us

$\displaystyle \frac{dy}{dx}=\sum_{j=0}^{\infty}(j+r)a_{j}x^{j+r-1}, (3)$

and

$\displaystyle \frac{d^{2}y}{dx^{2}}=\sum_{j=0}^{\infty}(j+r)(j+r-1)a_{j}x^{j+r-2}. (4)$

Substitution into Eq.(1) and noting the order of the equation we arrive at

$\displaystyle x^{2}\sum_{j=0}^{\infty}(j+r)(j+r-1)a_{j}x^{j+r-2}+x\sum_{j=0}^{\infty}(j+r)a_{j}x^{j+r-1}+x^{2}\sum_{j=0}^{\infty}a_{j}x^{j+r}=0. (5)$

Distribution and simplification of Eq.(5) yields

$\displaystyle \sum_{j=0}^{\infty}\bigg\{(j+r)(j+r-1)+(j+r)\bigg\}a_{j}x^{j+r}+\sum_{j=0}^{\infty}a_{j}x^{j+r+2}=0. (6)$

If we evaluate the terms in which $j=0$ and $j=1$, we get the following

$\displaystyle a_{0}\bigg\{r(r-1)+r\bigg\}x^{r}+a_{1}\bigg\{(1+r)r+(1+r)\bigg\}x^{r+1}+\sum_{j=2}^{\infty}\bigg\{[(j+r)(j+r-1)+(j+r)]a_{j}+a_{j-2}\bigg\}x^{j+r}=0, (7)$

where I have introduced the dummy variable $m=(j+r)-2$ and I have shifted the indices downward by 2. Consider now the indicial equation (coefficients of $a_{0}x^{r}$),

$\displaystyle r(r-1)+r=0, (8)$

which upon solving gives $r=r_{1}=r_{2}=0$. We may determine the recurrence relation from summation terms from which we get

$\displaystyle a_{j}(r)=\frac{-a_{j-2}(r)}{[(j+r)(j+r-1)+(j+r)]}=\frac{-a_{j-2}(r)}{(j+r)^{2}}. (9)$

To determine $J_{0}(x)$ we let $r=0$ in which case the recurrence relation becomes

$\displaystyle a_{j}=\frac{-a_{j-2}}{j^{2}}, (10)$

where $j=2,4,6,...$. Thus we have

$\displaystyle J_{0}(x)=a_{0}x^{0}+a_{1}x+... (11)$

The only way the second term above is 0 is if $a_{1}=0$. So, the successive terms are $a_{3},a_{5},a_{7},..., = 0$. Let $j=2k$, where $k\in \mathbb{Z}^{+}$, then the recurrence relation is again modified to

$\displaystyle a_{2k}=\frac{-a_{2k-2}}{(2k)^{2}}. (12)$

In general, for any value of $k$, one finds the expression

$\displaystyle ... \frac{(-1)^{k}a_{0}x^{2k}}{2^{2k}(k!)^{2}}. (13)$

Thus our solution for the Bessel function of the first kind is

$\displaystyle J_{0}(x)=a_{0}\bigg\{1+\sum_{k=1}^{\infty}\frac{(-1)^{k}x^{2k}}{2^{2k}(k!)^{2}}\bigg\}. (14)$

# Basics of Tensor Calculus and General Relativity-Vectors and Introduction to Tensors (Part II-Continuation of Vectors)

SOURCE FOR CONTENT: Neuenschwander, D.E., 2015. Tensor Calculus for Physics. Johns Hopkins University Press. Ch.1

In the preceding post of this series, we saw how we may define a vector in the traditional sense. There is another formulation which is the focus of this post. One becomes familiar with this formulation typically in a second course in quantum mechanics or of a similar form in an introductory linear algebra course.

A vector may be written in two forms: a ket vector,

$\displaystyle |A\rangle=\begin{pmatrix} a_{1}&\\ a_{2}&\\ \vdots &\\ a_{N} \end{pmatrix}, (1.1)$

or a bra vector (conjugate vector):

$\displaystyle \langle A|=\begin{pmatrix} a_{1} & a_{2} & \hdots & a_{N} \end{pmatrix}, (1.2)$

Additionally, if $\langle A| \in \mathbb{C}$, then the conjugate vector takes the form

$\displaystyle \langle A|=\begin{pmatrix} a_{1}^{*} & a_{2}^{*} & \hdots & a_{N}^{*} \end{pmatrix}. (1.3)$

In words, if the conjugate vector exists in the complex plane, we may express such a vector in terms of complex conjugates of the components.

We may form the inner product by the following

$\displaystyle \langle A|B\rangle = a_{1}^{*}b_{1}+a_{2}^{*}b_{2}+\hdots+a_{N}^{*}b_{M}=\sum_{i=1}^{N}\sum_{j=1}^{M}a_{i}^{*}b_{j}. (2)$

Conversely we may form the outer product as follows

$\displaystyle |A\rangle \langle B|= \begin{pmatrix} a_{1}b_{1}^{*} & a_{1}b_{2}^{*} & \hdots & a_{1}b_{M}^{*}\\ a_{2}b_{1}^{*} & a_{2}b_{2}^{*} & \hdots & a_{2}b_{M}^{*}\\ \vdots & \vdots & \ddots & \vdots \\ a_{N}b_{1}^{*} & a_{N}b_{2}^{*}& \hdots & a_{N}b_{M}^{*}\\ \end{pmatrix}, (3)$

Additionally, one typically defines a vector as a linear combination of basis vectors which we shall express by the following:

$\displaystyle \hat{i}=|1\rangle \equiv \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix},$

$\displaystyle \hat{j} = |2\rangle \equiv \begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix},$

$\displaystyle \hat{k}=|3\rangle \equiv \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix},$

where in general the basis vectors satisfy $\langle i|j \rangle = \delta_{ij}$. Therefore we can write any arbitrary vector in the following way

$\displaystyle \textbf{A}= a_{1}|1\rangle + a_{2}|2\rangle + ... + a_{n}|n\rangle. (4)$

Moreover, any conjugate vector may be written as

$\displaystyle \langle B|= b_{1}\langle 1| + b_{2}\langle 2| + ... + b_{m}\langle m|. (5)$

Let $\mathcal{P} = |A\rangle \langle B|$ represent the outer product which may also be written in component form as $\mathcal{P}_{ij}=\langle i|\mathcal{P}|j\rangle$. We therefore have the condition that

$\displaystyle \textbf{1} = \sum_{\mu}|\mu\rangle \langle \mu|, (6)$

and

$\displaystyle |A\rangle = \sum_{\mu}|\mu\rangle \langle \mu|A\rangle =\sum_{\mu}A^{\mu}|\mu\rangle, (7)$

wherein $\displaystyle A^{\mu}\equiv \langle \mu|A\rangle$. The first relation represents the completeness of an orthonormal set of basis vectors and the second is its modified form.

The next post will discuss the transformations of coordinates of vectors using both the matrix formulation and using partial derivatives.