We address the critical and universal aspects of counterion-condensationtransition at a single charged cylinder in both two and three spatialdimensions using numerical and analytical methods. By introducing a novelMonte-Carlo sampling method in logarithmic radial scale, we are able tonumerically simulate the critical limit of infinite system size (correspondingto infinite-dilution limit) within tractable equilibration times. The criticalexponents are determined for the inverse moments of the counterionic densityprofile (which play the role of the order parameters and represent the inverselocalization length of counterions) both within mean-field theory and withinMonte-Carlo simulations. In three dimensions (3D), correlation effects(neglected within mean-field theory) lead to an excessive accumulation ofcounterions near the charged cylinder below the critical temperature(condensation phase), while surprisingly, the critical region exhibitsuniversal critical exponents in accord with the mean-field theory. In twodimensions (2D), we demonstrate, using both numerical and analyticalapproaches, that the mean-field theory becomes exact at all temperatures(Manning parameters), when number of counterions tends to infinity. For finiteparticle number, however, the 2D problem displays a series of peculiar singularpoints (with diverging heat capacity), which reflect successive de-localizationevents of individual counterions from the central cylinder. In both 2D and 3D,the heat capacity shows a universal jump at the critical point, and the energydevelops a pronounced peak. The asymptotic behavior of the energy peak locationis used to locate the critical temperature, which is also found to be universaland in accordance with the mean-field prediction.
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