Let \pi(x) be a given probability density proportional to \exp(-U(x)) in a high-dimensional Euclidean space \mathbb{R}^m. The diffusion dX(t) = -\nabla U(X(t))dt + \sqrt 2 dW(t) is often used to sample from \pi. Instead of -\nabla U(x), we consider diffusions with smooth drift b(x) and having equilibrium \pi(x). First we study some general properties and then concentrate on the Gaussian case, namely, -\nabla U(x) = Dx with a strictly negative-definite real matrix D and b(x) = Bx with a stability matrix B; that is, the real parts of the eigenvalues of B are strictly negative. Using the rate of convergence of the covariance of X(t) [or together with EX(t)] as the criterion, we prove that, among all such b(x), the drift Dx is the worst choice and that improvement can be made if and only if the eigenvalues of D are not identical. In fact, the convergence rate of the covariance is \exp(2\lambda_M(B)t), where \lambda_M(B) is the maximum of the real parts of the eigenvalues of B and the infimum of \lambda_M(B) over all such B is 1/m \operatorname{tr} D. If, for example, a "circulant" drift \bigg(\frac{\partial U}{\partial x_m} - \frac{\partial U}{\partial x_2},\frac{\partial U}{\partial x_1} - \frac{\partial U}{\partial x_3}, \cdots, \frac{\partial U}{\partial x_{m-1}} - \frac{\partial U}{\partial x_1}\bigg) is added to Dx, then for essentially all D, the diffusion with this modified drift has a better convergence rate.