def pdf_Gates_Gaudin_Schuhman(d, d_characteristic, m):
    """Calculates the probability density of a particle
    distribution following the Gates, Gaudin and Schuhman (GGS) model given a 
    particle diameter `d`, characteristic (maximum) particle
    diameter `d_characteristic`, and exponent `m`.
    
    .. math::
        q(d) = \\frac{n}{d}\\left(\\frac{d}{d_{characteristic}}\\right)^m 
        \\text{ if } d < d_{characteristic} \\text{ else } 0
        
    Parameters
    ----------
    d : float
        Specified particle diameter, [m]
    d_characteristic : float
        Characteristic particle diameter; in this model, it is the largest 
        particle size diameter in the distribution, [m]
    m : float
        Particle size distribution exponent, [-]    

    Returns
    -------
    pdf : float
        GGS probability density function, [-]

    Notes
    -----
    The characteristic diameter can be in terns of number density (denoted 
    :math:`q_0(d)`), length density (:math:`q_1(d)`), surface area density
    (:math:`q_2(d)`), or volume density (:math:`q_3(d)`). Volume density is
    most often used. Interconversions among the distributions is possible but
    tricky.

    Examples
    --------
    >>> pdf_Gates_Gaudin_Schuhman(d=2E-4, d_characteristic=1E-3, m=2.3)
    283.8355768512045

    References
    ----------
    .. [1] Schuhmann, R., 1940. Principles of Comminution, I-Size Distribution 
       and Surface Calculations. American Institute of Mining, Metallurgical 
       and Petroleum Engineers Technical Publication 1189. Mining Technology, 
       volume 4, p. 1-11.
    .. [2] Bayat, Hossein, Mostafa Rastgo, Moharram Mansouri Zadeh, and Harry 
       Vereecken. "Particle Size Distribution Models, Their Characteristics and
       Fitting Capability." Journal of Hydrology 529 (October 1, 2015): 872-89.
    """
    if d <= d_characteristic:
        return m / d * (d / d_characteristic) ** m
    else:
        return 0.0
