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Debye entropic force and modified Newtonian dynamics

Verlinde has suggested that the gravity has an entropic origin, and a gravitational system could be regarded as a thermodynamical system. It is well-known that the equipartition law of energy is invalid at very low temperature. Therefore, entropic force should be modified while the temperature of the holographic screen is very low. It is shown that the modified entropic force is proportional to the square of the acceleration, while the temperature of the holographic screen is much lower than the Debye temperature $T_D$. The modified entropic force returns to the Newton's law of gravitation while the temperature of the holographic screen is much higher than the Debye temperature. The modified entropic force is connected with modified Newtonian dynamics (MOND). The constant $a_0$ involved in MOND is linear in the Debye frequency $ω_D$, which can be regarded as the largest frequency of the bits in screen. We find that there do have a strong connection between MOND and cosmology in the framework of Verlinde's entropic force, if the holographic screen is taken to be bound of the Universe. The Debye frequency is linear in the Hubble constant $H_0$.