Complex Langevin simulation of QCD at finite density and low temperature using the deformation technique

We study QCD at finite density and low temperature by using the complex Langevin method. We employ the gauge cooling to control the unitarity norm and introduce a deformation parameter in the Dirac operator to avoid the singular-drift problem. The reliability of the obtained results are judged by the probability distribution of the magnitude of the drift term. By making extrapolations with respect to the deformation parameter using only the reliable results, we obtain results for the original system. We perform simulations on a $4^3\times 3$ lattice and show that our method works well even in the region where the reweighting method fails due to the severe sign problem. As a result we observe a delayed onset of the baryon number density as compared with the phase-quenched model, which is a clear sign of the Silver Blaze phenomenon.Comment: 8 pages, 6 figures, presented at the 35th International Symposium on Lattice Field Theory (Lattice 2017), 18-24 June 2017, Granada, Spai

Testing the criterion for correct convergence in the complex Langevin method

Recently the complex Langevin method (CLM) has been attracting attention as a solution to the sign problem, which occurs in Monte Carlo calculations when the effective Boltzmann weight is not real positive. An undesirable feature of the method, however, was that it can happen in some parameter regions that the method yields wrong results even if the Langevin process reaches equilibrium without any problem. In our previous work, we proposed a practical criterion for correct convergence based on the probability distribution of the drift term that appears in the complex Langevin equation. Here we demonstrate the usefulness of this criterion in two solvable theories with many dynamical degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex coupling constant and the chiral Random Matrix Theory for finite density QCD, which were studied by the CLM before. Our criterion can indeed tell the parameter regions in which the CLM gives correct results.Comment: 16 pages, 2 figures; (v2) reference and comment added; (v3) minor revision; (v4) final version published in JHE

Concentration Gradient in a Continuous Countercurrent Extraction Column with Longitudinal Back-Mixing

The following relations have been derived by the authors for the variation of concentration in a continuous countercurrent extraction column in which there is longitudinal back-mixing of the continuous phase, on the assumption that the diffusivity of the back-mixing and liquid velocity are constant and the solutions are dilute : When 1+β>0 Cc-α/Cc₀-α = (1+√1+β)e²ᵞ√¹⁺ᵝ⁽¹⁻ˣ/ᴸ⁾-(1-√1+β)2√1+βeᵞ⁽¹⁺√¹⁺ᵝ⁾⁽¹⁻ˣ/ᴸ⁾ and when 1+β<0 Cc-α/Cc₀-α = i√1+β cos[γi√1+β(1-x/L)]+sin[γi√1+β(1-x/L)]i√1+βeᵞ⁽¹⁻ˣ/ᴸ⁾ where : α = C*c₀-(mFc/Fd)Cc₀1-(mFc/Fd), β = 2MN/(M+NQ)², γ = M+NQ, M = Luc/2EcHc, N = Kcα′(1- mFc/Fd)(LHc/uc), Q = 1/2 1/(Fd/mFc)-1. The applicability of these equations is discussed, and the performance of an extraction column is analyzed

Criteria for the Scaling Up of Mixing Vessels

There are several concepts for the scaling up of mixing vessels. They are neither consistent nor conclusive. The authors propose an idea that the selection must be made depending upon the mixing objects. The classification of the types of scaling up is as follows : (1) Similarity for power requirement. Power requirement is correlated by Reynolds-, Froude- and Weber-numbers. The latter two are negligible in an ordinary correlation. (2) Similarity for mixing velocity of homogeneous liquid phase. The authors conclude that the similarity in blending speed is obtained by equal impeller speeds. (3) Similarity for heat and mass transfer on the fixed surface. Rushton et al. proposed the method of scaling-up by the following equation. (n₂/n₁)=(D₁/D₂)⁽²ˣ⁻¹⁾/ˣ This criterion should be limited in the case of the heat and mass transfer on a fixed surface. (4) Similarity for suspension of solid particles, dispersion of gas and liquid, and mass transfer on dispersed objects. The authors support the criterion of equal power per unit volume proposed by W.Büche