# 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.