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We present a model of pressure effects of a two-band superconductor based ona Ginzburg-Landau free energy with two order parameters. The parameters of thetheory are pressure as well as temperature dependent. New pressure effectsemerge as a result of the competition between the two bands. The theory then isapplied to MgB2. We identify two possible scenaria regarding the fate of thetwo $\sigma$ subbands under pressure, depending on whether or not both subbandsare above the Fermi energy at ambient pressure. The splitting of the twosubbands is probably caused by the E2g distortion. If only one subband is abovethe Fermi energy at ambient pressure (scenario I), application of pressurediminishes the splitting and it is possible that the lower subband participatesin the superconductivity. The corresponding crossover pressure and Gruneisenparameter are estimated. In the second scenario both bands start above theFermi energy and they move below it, either by pressure or via the substitutionof Mg by Al. In both scenaria, the possibility of electronical topologicaltransition is emphasized. Experimental signatures of both scenaria arepresented and existing experiments are discussed in the light of the differentphysical pictures.Comment: 6 pages; supersedes the first part of cond-mat/0204085 due to new  experiment

Ginzburg-Landau theory and effects of pressure on a two-band superconductor: application to MgB $\scriptstyle \mathsf {}$ 2