Stochastic Theories and Deterministic Differential Equations

Figure 3

Quantum and Liouville solutions without friction as a function of space x and time t. 𝜌^𝑄_𝑋(𝑡₀,𝑥)=𝜌^𝐿_𝑋(𝑡₀,𝑥)=(2𝜋𝑎²)^−1/2𝑒^{−(𝑥−𝑥0)2/(2𝑎2)},𝜌^𝐿_𝑌(𝑡₀,𝑦)=(2𝜋𝑏²)^−1/2𝑒^{−𝑦2/(2𝑏2)},𝑏=1/(2𝑎),𝑎²=1/2,𝑥₀=1.5, and 𝜀=0.