conspire/physics/molecular/single_chain/fjc/
mod.rs

1#[cfg(test)]
2mod test;
3
4use crate::{
5    math::{
6        Scalar, TensorArray, random_uniform,
7        special::{inverse_langevin, langevin, langevin_derivative, sinhc},
8    },
9    mechanics::CurrentCoordinate,
10    physics::molecular::single_chain::{
11        Configuration, Ensemble, Inextensible, Isometric, Isotensional, Legendre, MonteCarlo,
12        SingleChain, SingleChainError, Thermodynamics,
13    },
14};
15use std::f64::consts::{PI, TAU};
16
17/// The freely-jointed chain model.
18#[derive(Clone, Debug)]
19pub struct FreelyJointedChain {
20    /// The link length $`\ell_b`$.
21    pub link_length: Scalar,
22    /// The number of links $`N_b`$.
23    pub number_of_links: u8,
24    /// The thermodynamic ensemble.
25    pub ensemble: Ensemble,
26}
27
28impl SingleChain for FreelyJointedChain {
29    fn link_length(&self) -> Scalar {
30        self.link_length
31    }
32    fn number_of_links(&self) -> u8 {
33        self.number_of_links
34    }
35}
36
37impl Inextensible for FreelyJointedChain {
38    /// ```math
39    /// \lim_{\eta\to\infty}\gamma(\eta) = 1
40    /// ```
41    fn maximum_nondimensional_extension(&self) -> Scalar {
42        1.0
43    }
44}
45
46impl Thermodynamics for FreelyJointedChain {
47    fn ensemble(&self) -> Ensemble {
48        self.ensemble
49    }
50}
51
52impl Isometric for FreelyJointedChain {
53    fn nondimensional_helmholtz_free_energy(
54        &self,
55        nondimensional_extension: Scalar,
56    ) -> Result<Scalar, SingleChainError> {
57        self.nondimensional_extension_check(nondimensional_extension)?;
58        if nondimensional_extension == 0.0 {
59            Ok(0.0)
60        } else {
61            let [s0, _, _] = treloar_sums(self.number_of_links(), nondimensional_extension);
62            Ok(nondimensional_extension.abs().ln() - s0.ln())
63        }
64    }
65    /// ```math
66    /// \eta(\gamma) = \frac{1}{N_b\gamma} + \left(\frac{1}{2} - \frac{1}{N_b}\right)\frac{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 3}}{\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2}}
67    /// ```
68    fn nondimensional_force(
69        &self,
70        nondimensional_extension: Scalar,
71    ) -> Result<Scalar, SingleChainError> {
72        self.nondimensional_extension_check(nondimensional_extension)?;
73        if nondimensional_extension == 0.0 {
74            Ok(0.0)
75        } else {
76            let [s0, s1, _] = treloar_sums(self.number_of_links(), nondimensional_extension);
77            let n = self.number_of_links() as Scalar;
78            Ok((1.0 / nondimensional_extension + (0.5 * n - 1.0) * s1 / s0) / n)
79        }
80    }
81    /// ```math
82    /// \kappa(\gamma) = \frac{\partial\eta}{\partial\gamma}
83    /// ```
84    fn nondimensional_stiffness(
85        &self,
86        nondimensional_extension: Scalar,
87    ) -> Result<Scalar, SingleChainError> {
88        self.nondimensional_extension_check(nondimensional_extension)?;
89        if nondimensional_extension == 0.0 {
90            Ok(Scalar::NAN)
91        } else {
92            let [s0, s1, s2] = treloar_sums(self.number_of_links(), nondimensional_extension);
93
94            if !s0.is_finite() || s0 == 0.0 {
95                return Ok(Scalar::NAN);
96            }
97
98            let n = self.number_of_links() as Scalar;
99            let p = n - 2.0;
100            let b = (0.5 * n - 1.0) / n;
101
102            let ds0dx = -(p / 2.0) * s1;
103            let ds1dx = -((p - 1.0) / 2.0) * s2;
104            let d_ratio_dx = (ds1dx * s0 - s1 * ds0dx) / (s0 * s0);
105
106            Ok(-1.0 / (n * nondimensional_extension * nondimensional_extension) + b * d_ratio_dx)
107        }
108    }
109    /// ```math
110    /// \mathcal{P}(\gamma) = \frac{1}{8\pi\gamma}\frac{N_b^{N_b}}{(N_b - 2)!}\sum_{s=0}^{s_\mathrm{max}}(-1)^s\binom{N_b}{s}\left(m - \frac{s}{N_b}\right)^{N_b - 2}
111    /// ```
112    fn nondimensional_spherical_distribution(
113        &self,
114        nondimensional_extension: Scalar,
115    ) -> Result<Scalar, SingleChainError> {
116        self.nondimensional_extension_check(nondimensional_extension)?;
117        if nondimensional_extension <= 0.0 || nondimensional_extension >= 1.0 {
118            Ok(0.0)
119        } else {
120            let number_of_links = self.number_of_links();
121            let [s0, _, _] = treloar_sums(number_of_links, nondimensional_extension);
122            let n = number_of_links as Scalar;
123            let factorial_n_minus_2 = (1..=(number_of_links - 2))
124                .map(|i| i as Scalar)
125                .product::<Scalar>();
126            Ok((n.powf(n) / (8.0 * PI * nondimensional_extension * factorial_n_minus_2)) * s0)
127        }
128    }
129}
130
131impl Isotensional for FreelyJointedChain {
132    /// ```math
133    /// \varrho(\eta) = N_b\ln\left[\frac{\eta}{\sinh(\eta)}\right]
134    /// ```
135    fn nondimensional_gibbs_free_energy_per_link(
136        &self,
137        nondimensional_force: Scalar,
138    ) -> Result<Scalar, SingleChainError> {
139        Ok(-sinhc(nondimensional_force).ln())
140    }
141    /// ```math
142    /// \gamma(\eta) = \mathcal{L}(\eta)
143    /// ```
144    fn nondimensional_extension(
145        &self,
146        nondimensional_force: Scalar,
147    ) -> Result<Scalar, SingleChainError> {
148        Ok(langevin(nondimensional_force))
149    }
150    /// ```math
151    /// \zeta(\eta) = \mathcal{L}'(\eta)
152    /// ```
153    fn nondimensional_compliance(
154        &self,
155        nondimensional_force: Scalar,
156    ) -> Result<Scalar, SingleChainError> {
157        Ok(langevin_derivative(nondimensional_force))
158    }
159}
160
161impl Legendre for FreelyJointedChain {
162    /// ```math
163    /// \eta(\gamma) = \mathcal{L}^{-1}(\gamma)
164    /// ```
165    fn nondimensional_force(
166        &self,
167        nondimensional_extension: Scalar,
168    ) -> Result<Scalar, SingleChainError> {
169        self.nondimensional_extension_check(nondimensional_extension)?;
170        Ok(inverse_langevin(nondimensional_extension))
171    }
172    /// ```math
173    /// \mathcal{P}(\gamma) \propto \left\{\frac{\sinh[\eta(\gamma)]}{\eta(\gamma)\exp[\eta(\gamma)\gamma]}\right\}^{N_b}
174    /// ```
175    fn nondimensional_spherical_distribution(
176        &self,
177        nondimensional_extension: Scalar,
178    ) -> Result<Scalar, SingleChainError> {
179        let nondimensional_force = Legendre::nondimensional_force(self, nondimensional_extension)?;
180        Ok(
181            (((nondimensional_force * (1.0 - nondimensional_extension)).exp()
182                - (-nondimensional_force * (1.0 + nondimensional_extension)).exp())
183                / 2.0
184                / nondimensional_force)
185                .powi(self.number_of_links() as i32)
186                / normalization(self.number_of_links()),
187        )
188    }
189}
190
191impl MonteCarlo for FreelyJointedChain {
192    fn random_configuration(&self) -> Configuration {
193        let mut position = CurrentCoordinate::zero();
194        (0..self.number_of_links())
195            .map(|_| {
196                let cos_theta = 2.0 * random_uniform() - 1.0;
197                let sin_theta = (1.0 - cos_theta * cos_theta).sqrt();
198                let phi = TAU * random_uniform();
199                let (sin_phi, cos_phi) = phi.sin_cos();
200                position[0] += sin_theta * cos_phi;
201                position[1] += sin_theta * sin_phi;
202                position[2] += cos_theta;
203                position.clone()
204            })
205            .collect()
206    }
207}
208
209fn treloar_sums(number_of_links: u8, x: Scalar) -> [Scalar; 3] {
210    if number_of_links <= 2 {
211        return [Scalar::NAN; 3];
212    }
213    let n = number_of_links as Scalar;
214    let p = (number_of_links - 2) as i32;
215    let m = 0.5 * (1.0 - x);
216    let k = ((n * m).ceil() as usize)
217        .saturating_sub(1)
218        .min(number_of_links as usize);
219    let k_float = n * m;
220    if (k_float - k_float.round()).abs() == 0.0 {
221        return [Scalar::NAN; 3];
222    }
223    let mut binom = 1.0;
224    let mut s0 = 0.0;
225    let mut s1 = 0.0;
226    let mut s2 = 0.0;
227    for s in 0..=k {
228        let sign = if s % 2 == 0 { 1.0 } else { -1.0 };
229        let t = m - (s as Scalar) / n;
230        let t0 = if p >= 0 {
231            t.powi(p)
232        } else if t == 0.0 {
233            0.0
234        } else {
235            t.powi(p)
236        };
237        let t1 = if p > 0 {
238            t.powi(p - 1)
239        } else if t == 0.0 {
240            0.0
241        } else {
242            t.powi(p - 1)
243        };
244        let t2 = if p > 1 {
245            t.powi(p - 2)
246        } else if t == 0.0 {
247            0.0
248        } else {
249            t.powi(p - 2)
250        };
251        s0 += sign * binom * t0;
252        s1 += sign * binom * t1;
253        s2 += sign * binom * t2;
254        let sf = s as Scalar;
255        binom *= (n - sf) / (sf + 1.0);
256    }
257    [s0, s1, s2]
258}
259
260fn normalization(number_of_links: u8) -> Scalar {
261    match number_of_links {
262        0 => Scalar::NAN,
263        1 => 1.389_063_303_837_301_3,
264        2 => 0.714_480_944_477_587_6,
265        3 => 0.446_182_225_454_993_8,
266        4 => 0.310_582_574_239_989_03,
267        5 => 0.231_583_731_936_937_35,
268        6 => 0.181_026_390_997_248_38,
269        7 => 0.146_444_713_993_307_1,
270        8 => 0.121_590_329_098_661_26,
271        9 => 0.103_031_548_251_807_95,
272        10 => 0.088_746_746_615_799_2,
273        11 => 0.077_477_021_054_147_71,
274        12 => 0.068_402_348_281_970_37,
275        13 => 0.060_968_329_153_341_53,
276        14 => 0.054_788_235_109_506_34,
277        15 => 0.049_585_008_268_986_38,
278        16 => 0.045_155_543_187_723_454,
279        17 => 0.041_347_934_607_350_624,
280        18 => 0.038_046_552_454_682_33,
281        19 => 0.035_161_995_757_729_23,
282        20 => 0.032_624_174_659_916_245,
283        21 => 0.030_377_448_781_842_47,
284        22 => 0.028_377_147_909_992_65,
285        23 => 0.026_587_040_753_179_49,
286        24 => 0.024_977_465_826_024_475,
287        25 => 0.023_523_932_434_435_773,
288        26 => 0.022_206_060_476_346_62,
289        27 => 0.021_006_767_816_333_316,
290        28 => 0.019_911_640_864_367_884,
291        29 => 0.018_908_442_314_802_338,
292        30 => 0.017_986_722_687_273_728,
293        31 => 0.017_137_511_214_426_318,
294        32 => 0.016_353_067_950_360_97,
295        33 => 0.015_626_683_526_651_468,
296        34 => 0.014_952_516_294_544_284,
297        35 => 0.014_325_459_026_026_574,
298        36 => 0.013_741_029_152_869_741,
299        37 => 0.013_195_277_875_651_702,
300        38 => 0.012_684_714_496_711_45,
301        39 => 0.012_206_243_109_212_051,
302        40 => 0.011_757_109_371_641_886,
303        41 => 0.011_334_855_558_612_255,
304        42 => 0.010_937_282_437_959_286,
305        43 => 0.010_562_416_805_450_496,
306        44 => 0.010_208_483_730_069_894,
307        45 => 0.009_873_882_738_568_507,
308        46 => 0.009_557_167_308_025_053,
309        47 => 0.009_257_027_147_393_918,
310        48 => 0.008_972_272_839_407_26,
311        49 => 0.008_701_822_487_349_997,
312        50 => 0.008_444_690_070_700_62,
313        51 => 0.008_199_975_262_200_628,
314        52 => 0.007_966_854_498_747_187,
315        53 => 0.007_744_573_131_302_788,
316        54 => 0.007_532_438_506_130_219,
317        55 => 0.007_329_813_852_159_101,
318        56 => 0.007_136_112_868_025_883,
319        57 => 0.006_950_794_917_986_018,
320        58 => 0.006_773_360_759_023_209,
321        59 => 0.006_603_348_732_523_006,
322        60 => 0.006_440_331_363_193_850_5,
323        61 => 0.006_283_912_315_803_12,
324        62 => 0.006_133_723_666_987_055,
325        63 => 0.005_989_423_455_088_637,
326        64 => 0.005_850_693_475_837_429,
327        65 => 0.005_717_237_295_843_889,
328        66 => 0.005_588_778_459_447_456,
329        67 => 0.005_465_058_867_524_899,
330        68 => 0.005_345_837_309_508_891,
331        69 => 0.005_230_888_132_150_507,
332        70 => 0.005_120_000_030_536_583,
333        71 => 0.005_012_974_948_588_448,
334        72 => 0.004_909_627_077_760_272,
335        73 => 0.004_809_781_943_954_87,
336        74 => 0.004_713_275_573_809_406_5,
337        75 => 0.004_619_953_732_495_746,
338        76 => 0.004_529_671_226_049_816,
339        77 => 0.004_442_291_262_007_673_5,
340        78 => 0.004_357_684_862_797_343,
341        79 => 0.004_275_730_326_926_852,
342        80 => 0.004_196_312_733_530_766,
343        81 => 0.004_119_323_486_298_74,
344        82 => 0.004_044_659_893_217_922,
345        83 => 0.003_972_224_778_923_046,
346        84 => 0.003_901_926_126_769_502,
347        85 => 0.003_833_676_748_030_511_5,
348        86 => 0.003_767_393_975_874_093_7,
349        87 => 0.003_702_999_382_002_534_4,
350        88 => 0.003_640_418_514_039_760_3,
351        89 => 0.003_579_580_651_933_304,
352        90 => 0.003_520_418_581_799_841,
353        91 => 0.003_462_868_385_788_78,
354        92 => 0.003_406_869_246_668_994,
355        93 => 0.003_352_363_265_961_134_8,
356        94 => 0.003_299_295_294_543_597_2,
357        95 => 0.003_247_612_774_755_336_3,
358        96 => 0.003_197_265_593_104_502_5,
359        97 => 0.003_148_205_942_769_372,
360        98 => 0.003_100_388_195_147_996,
361        99 => 0.003_053_768_779_776_389_4,
362        100 => 0.003_008_306_071_992_423_4,
363        101 => 0.002_963_960_287_774_609_6,
364        102 => 0.002_920_693_385_232_166,
365        103 => 0.002_878_468_972_265_675,
366        104 => 0.002_837_252_219_956_577_4,
367        105 => 0.002_797_009_781_279_318,
368        106 => 0.002_757_709_714_762_243_4,
369        107 => 0.002_719_321_412_752_858,
370        108 => 0.002_681_815_533_969_964_8,
371        109 => 0.002_645_163_940_049_785,
372        110 => 0.002_609_339_635_815_636,
373        111 => 0.002_574_316_713_021_305,
374        112 => 0.002_540_070_297_337_108_2,
375        113 => 0.002_506_576_498_364_846,
376        114 => 0.002_473_812_362_483_738,
377        115 => 0.002_441_755_828_343_935,
378        116 => 0.002_410_385_684_837_54,
379        117 => 0.002_379_681_531_389_384_6,
380        118 => 0.002_349_623_740_421_053,
381        119 => 0.002_320_193_421_852_075,
382        120 => 0.002_291_372_389_511_768,
383        121 => 0.002_263_143_129_344_052_4,
384        122 => 0.002_235_488_769_295_701_7,
385        123 => 0.002_208_393_050_786_021,
386        124 => 0.002_181_840_301_662_884_3,
387        125 => 0.002_155_815_410_556_500_4,
388        126 => 0.002_130_303_802_548_212_4,
389        127 => 0.002_105_291_416_077_128,
390        128 => 0.002_080_764_681_012_503,
391        129 => 0.002_056_710_497_824_479_8,
392        130 => 0.002_033_116_217_790_202_3,
393        131 => 0.002_009_969_624_176_372,
394        132 => 0.001_987_258_914_343_074,
395        133 => 0.001_964_972_682_717_241_4,
396        134 => 0.001_943_099_904_587_340_1,
397        135 => 0.001_921_629_920_673_915_4,
398        136 => 0.001_900_552_422_433_467_6,
399        137 => 0.001_879_857_438_055_702_6,
400        138 => 0.001_859_535_319_116_718_1,
401        139 => 0.001_839_576_727_852_897_4,
402        140 => 0.001_819_972_625_022_462_4,
403        141 => 0.001_800_714_258_323_577,
404        142 => 0.001_781_793_151_339_777_3,
405        143 => 0.001_763_201_092_985_212,
406        144 => 0.001_744_930_127_423_801_3,
407        145 => 0.001_726_972_544_437_928,
408        146 => 0.001_709_320_870_223_689,
409        147 => 0.001_691_967_858_591_060_8,
410        148 => 0.001_674_906_482_548_559_4,
411        149 => 0.001_658_129_926_253_140_5,
412        150 => 0.001_641_631_577_307_177_5,
413        151 => 0.001_625_405_019_385_355_6,
414        152 => 0.001_609_444_025_175_287_7,
415        153 => 0.001_593_742_549_616_558,
416        154 => 0.001_578_294_723_423_718_3,
417        155 => 0.001_563_094_846_879_572_4,
418        156 => 0.001_548_137_383_885_819,
419        157 => 0.001_533_416_956_258_811_4,
420        158 => 0.001_518_928_338_258_862_9,
421        159 => 0.001_504_666_451_342_125,
422        160 => 0.001_490_626_359_124_665_4,
423        161 => 0.001_476_803_262_548_898,
424        162 => 0.001_463_192_495_243_045,
425        163 => 0.001_449_789_519_064_795,
426        164 => 0.001_436_589_919_820_758_2,
427        165 => 0.001_423_589_403_153_785_4,
428        166 => 0.001_410_783_790_590_583,
429        167 => 0.001_398_169_015_742_466_7,
430        168 => 0.001_385_741_120_652_447_7,
431        169 => 0.001_373_496_252_282_184_2,
432        170 => 0.001_361_430_659_132_658_5,
433        171 => 0.001_349_540_687_992_734_2,
434        172 => 0.001_337_822_780_810_052_7,
435        173 => 0.001_326_273_471_678_944_4,
436        174 => 0.001_314_889_383_940_535_3,
437        175 => 0.001_303_667_227_389_772_3,
438        176 => 0.001_292_603_795_585_479_2,
439        177 => 0.001_281_695_963_258_609_4,
440        178 => 0.001_270_940_683_814_773_4,
441        179 => 0.001_260_334_986_927_068_6,
442        180 => 0.001_249_875_976_215_474_5,
443        181 => 0.001_239_560_827_009_231_7,
444        182 => 0.001_229_386_784_188_803_6,
445        183 => 0.001_219_351_160_104_175_2,
446        184 => 0.001_209_451_332_566_384_2,
447        185 => 0.001_199_684_742_909_333_8,
448        186 => 0.001_190_048_894_119_060_9,
449        187 => 0.001_180_541_349_027_766_8,
450        188 => 0.001_171_159_728_570_035,
451        189 => 0.001_161_901_710_098_784_4,
452        190 => 0.001_152_765_025_758_600_2,
453        191 => 0.001_143_747_460_914_206_6,
454        192 => 0.001_134_846_852_631_931_8,
455        193 => 0.001_126_061_088_212_113_2,
456        194 => 0.001_117_388_103_770_484_4,
457        195 => 0.001_108_825_882_866_665,
458        196 => 0.001_100_372_455_177_960_5,
459        197 => 0.001_092_025_895_216_747_5,
460        198 => 0.001_083_784_321_089_808_8,
461        199 => 0.001_075_645_893_298_036_3,
462        200 => 0.001_067_608_813_574_995_4,
463        201 => 0.001_059_671_323_762_909_2,
464        202 => 0.001_051_831_704_724_672_7,
465        203 => 0.001_044_088_275_290_578_8,
466        204 => 0.001_036_439_391_238_473_6,
467        205 => 0.001_028_883_444_306_137,
468        206 => 0.001_021_418_861_234_703,
469        207 => 0.001_014_044_102_842_014_4,
470        208 => 0.001_006_757_663_124_827_5,
471        209 => 0.000_999_558_068_388_836_5,
472        210 => 0.000_992_443_876_405_530_2,
473        211 => 0.000_985_413_675_594_929_3,
474        212 => 0.000_978_466_084_233_29,
475        213 => 0.000_971_599_749_684_902_6,
476        214 => 0.000_964_813_347_657_139_1,
477        215 => 0.000_958_105_581_477_943_6,
478        216 => 0.000_951_475_181_394_988_1,
479        217 => 0.000_944_920_903_895_750_9,
480        218 => 0.000_938_441_531_047_793_8,
481        219 => 0.000_932_035_869_858_555_4,
482        220 => 0.000_925_702_751_653_993_6,
483        221 => 0.000_919_441_031_475_440_3,
484        222 => 0.000_913_249_587_494_056_1,
485        223 => 0.000_907_127_320_442_295_6,
486        224 => 0.000_901_073_153_061_812_5,
487        225 => 0.000_895_086_029_567_262_1,
488        226 => 0.000_889_164_915_125_473_2,
489        227 => 0.000_883_308_795_349_485_9,
490        228 => 0.000_877_516_675_806_960_8,
491        229 => 0.000_871_787_581_542_502_7,
492        230 => 0.000_866_120_556_613_433,
493        231 => 0.000_860_514_663_638_586_4,
494        232 => 0.000_854_968_983_359_706_2,
495        233 => 0.000_849_482_614_215_034_8,
496        234 => 0.000_844_054_671_924_712_5,
497        235 => 0.000_838_684_289_087_606_4,
498        236 => 0.000_833_370_614_789_207_2,
499        237 => 0.000_828_112_814_220_247_1,
500        238 => 0.000_822_910_068_305_699,
501        239 => 0.000_817_761_573_343_835,
502        240 => 0.000_812_666_540_655_029_1,
503        241 => 0.000_807_624_196_240_001_8,
504        242 => 0.000_802_633_780_447_217_2,
505        243 => 0.000_797_694_547_649_146_8,
506        244 => 0.000_792_805_765_927_132_3,
507        245 => 0.000_787_966_716_764_583_6,
508        246 => 0.000_783_176_694_748_257,
509        247 => 0.000_778_435_007_277_372_5,
510        248 => 0.000_773_740_974_280_329_3,
511        249 => 0.000_769_093_927_938_796_6,
512        250 => 0.000_764_493_212_418_953_9,
513        251 => 0.000_759_938_183_609_671_9,
514        252 => 0.000_755_428_208_867_465_4,
515        253 => 0.000_750_962_666_767_783,
516        254 => 0.000_746_540_946_863_032_5,
517        255 => 0.000_742_162_449_446_367_6,
518    }
519}
conspire/physics/molecular/single_chain/fjc/mod.rs

conspire/physics/molecular/single_chain/fjc/
mod.rs
