text: The freely joined chain, also called an ideal chain, follows the random walk model. Microscopically, the 3-D random walk of a polymer chain assumes the overall end-to-end distance is expressed in terms of the x, y and z directions:
  In the model,  is the length of a rigid segment,  is the number of segments of length ,  is the distance between the fixed and free ends, and  is the "contour length" or . Above the glass transition temperature, the polymer chain oscillates and  changes over time. The probability distribution of the chain is the product of the probability distributions of the individual components, given by the following Gaussian distribution:
  Therefore, the ensemble average end-to-end distance is simply the standard integral of the probability distribution over all space. Note that the movement could be backwards or forwards, so the net average  will be zero. However, one can use the root mean square as a useful measure of the distance.
  The Flory theory of rubber elasticity has pointed out the rubber elasticity has primarily entropic origins. By using the following basic equations for Helmholtz free energy and its discussion about entropy, the force generated from the deformation of a rubber chain from its original un-stretched conformation can be derived. The  is the number of conformations of the polymer chain. Since the deformation does not involve enthalpy change, the change in free energy can just be calculated as the change in entropy. It can be observed that the force equation resembles the behavior of a spring and follows Hookes law: , where F is the force, k is the spring constant and x' is the distance. Usually, neo-Hookean model can be used on cross-linked polymers to predict their stress-strain relations:
  Note that the elastic coefficient  is temperature dependent. If we increase the rubber temperature, the elastic coefficient also rises. This is the reason why rubber under constant stress shrinks when its temperature increases.
  We can further expand the Flory theory into a macroscopic view, where bulk rubber material is discussed. Assume the original dimension of the rubber material is ,  and , a deformed shape can then be expressed by applying an individual extension ratio  to the length (, , ). So microscopically, the deformed polymer chain can also be expressed with the extension ratio: , , . The free energy change due to deformation can then be expressed as follows:
  Assume that the rubber is cross-linked and isotropic, the random walk model gives ,  and  are distributed according to a normal distribution. Therefore, they are equal in space, and all of them are 1/3 of the overall end-to-end distance of the chain: . Plugging in the change of free energy equation above, it is easy to get:
  The free energy change per volume is just:
  where  is the number of strands in network, the subscript "def" means "deformation", , which is the number density per volume of polymer chains,  which is the ratio between the end-to-end distance of the chain and the theoretical distance that obey random walk statistics.  If we assume incompressibility, the product of extension ratios is 1, implying no change in the volume: .
  In a uniaxial deformed rubber, because  we assume . So the previous free energy per volume equation is:
  The engineering stress (by definition) is the first derivative of the energy in terms of the extension ratio, which is equivalent to the concept of strain:
  and the Young's Modulus  is defined as derivative of the stress with respect to strain, which measures the stiffness of the rubber in laboratory experiments.
  <math display="block">E=\frac{d(\sigma_\text{eng})}{d\lambda_z}=k_\text{B}T v_s \beta \left.\left(1+ \frac{2}{\lambda_z^3}\right)\right|_{\lambda_z=1}
  = 3 k_\text{B}T v_s\beta = \frac{3\rho \beta RT}{M_s},  is the mass density of the chain,  is the number average molecular weight of a network strand between crosslinks. Here, this type of analysis links the thermodynamic theory of rubber elasticity to experimentally measurable parameters. In addition, it gives in sights into the cross-linking condition of the materials.