Math 2.0

Statistical mechanics, structure of matter Polymers and soft matter Example

Ideal chain

Builds on Random walkCentral limit theoremNormal distribution

Example

An ideal chain, or freely jointed chain, models a polymer of NN segments of fixed length b>0b > 0 whose directions are independent. Let u1,…,uNu_1, \dots, u_N be independent random unit vectors in ℝ3\R^3, each uniformly distributed on the unit sphere. The end-to-end vector is R=b∑i=1NuiR = b \sum_{i=1}^N u_i, so the chain is a random walk in ℝ3\R^3 with steps of length bb. Check of the mean square size: 𝔼[ui]=0\mathbb{E}[u_i] = 0 and 𝔼[ui⋅uj]=0\mathbb{E}[u_i \cdot u_j] = 0 for i≠ji \ne j by independence, while ui⋅ui=1u_i \cdot u_i = 1. So 𝔼[R]=0\mathbb{E}[R] = 0 and 𝔼[∣R∣2]=b2∑i,j𝔼[ui⋅uj]=Nb2.\mathbb{E}\big[\abs{R}^2\big] = b^2 \sum_{i, j} \mathbb{E}[u_i \cdot u_j] = N b^2. The typical size Nb\sqrt{N} b is much smaller than the contour length NbN b.

Remarks

Each coordinate of uiu_i has mean 00 and variance 1/31/3. By the central limit theorem, for large NN the vector RR is close to a normal distribution with density P(R)=(32πNb2)3/2exp⁡(−3∣R∣22Nb2).P(R) = \Big( \frac{3}{2 \pi N b^2} \Big)^{3/2} \exp\Big( -\frac{3 \abs{R}^2}{2 N b^2} \Big). At temperature TT the entropy kBln⁡P(R)k_B \ln P(R) gives the free energy F(R)=3kBT2Nb2∣R∣2F(R) = \frac{3 k_B T}{2 N b^2} \abs{R}^2 up to a constant. So an ideal chain behaves as a spring of stiffness 3kBT/(Nb2)3 k_B T / (N b^2), of purely entropic origin. The model ignores the fact that two segments cannot overlap; a real chain in a good solvent is better described by a self-avoiding walk.

Reading path

To understand this page from the ground up, read these 70 pages in order. The foundations come first.

  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Normal distribution concept
  6. Ordered field definition
  7. Proposition concept
  8. Random experiment concept
  9. Random walk concept
  10. Rational numbers object
  11. Real numbers object
  12. Euler's formula statement
  13. Set definition
  14. Ring of integers statement
  15. Disjunction definition
  16. Implication definition
  17. Predicate definition
  18. Ordered pair definition
  19. Union definition
  20. Sample space definition
  21. Existential quantifier definition
  22. Universal quantifier definition
  23. The rationals form a field statement
  24. Cartesian product definition
  25. Subset definition
  26. Uniqueness quantifier definition
  27. The rationals form an ordered field statement
  28. Binary relation definition
  29. Power set definition
  30. Universal set definition
  31. Order on the reals definition
  32. Function definition
  33. Complement definition
  34. Interval definition
  35. Injective function definition
  36. Indicator function definition
  37. Preimage of a set definition
  38. Sigma-algebra definition
  39. Countable set definition
  40. Event definition
  41. Borel sigma-algebra definition
  42. Probability axioms concept
  43. Probability measure definition
  44. Probability distribution definition
  45. Dirac measure definition
  46. Probability space definition
  47. Conditional probability definition
  48. Random variable definition
  49. Cumulative distribution function definition
  50. Discrete random variable definition
  51. Law of a random variable definition
  52. Independence of two events definition
  53. Random variable with a density definition
  54. Convergence in distribution definition
  55. Probability mass function definition
  56. Expected value definition
  57. Identically distributed random variables definition
  58. Mutual independence of events definition
  59. Probability density function definition
  60. Law of the unconscious statistician statement
  61. Independent sigma-algebras definition
  62. Density of the normal distribution definition
  63. Characteristic function definition
  64. Independence of random variables definition
  65. Normal distribution definition
  66. The characteristic function determines the distribution statement
  67. Independent and identically distributed random variables definition
  68. Lévy's continuity theorem statement
  69. Central limit theorem statement
  70. Ideal chain example

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ideal_chain
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TagM5WZKH7ZDV
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MSC82D60, 60G50
Instance ofRandom walk
UsesRandom walk, Central limit theorem, Normal distribution
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Changed3 November 2026

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