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 C_V = 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.

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