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utils

detailed_balance

detailed_balance_factor(energy, temperature, energy_unit='meV', temperature_unit='K', divide_by_temperature=True)

Compute the detailed balance factor (DBF): $$ DBF(E, T) = E(n(E)+1)=\frac{E}{(1 - e^{-E / (k_B*T)})}}, $$ where \(n(E)\) is the Bose-Einstein distribution, \(E\) is the energy transfer, and \(T\) is the temperature. \(k_B\) is the Boltzmann constant. If divide_by_temperature is True, the result is normalized by \(k_B*T\) to have value 1 at \(E=0\).

Parameters:

Name Type Description Default
energy int | float | list | ndarray | Variable

The energy transfer. If number, assumed to be in meV unless energy_unit is set.

 required
temperature int | float | Variable | Parameter

The temperature. If number, assumed to be in K unless temperature_unit is set.

 required
energy_unit str | sc.Unit, default='meV'

Unit for energy if energy is given as a number or list. Default is 'meV'

 'meV'
temperature_unit str | sc.Unit, default='K'

Unit for temperature if temperature is given as a number. Default is 'K'

 'K'
divide_by_temperature bool, default=True

If True, divide the result by \(k_B*T\) to make it dimensionless and have value 1 at E=0. Default is True.

 True

Returns:

Type Description
ndarray

np.ndarray: Detailed balance factor evaluated at the given energy and temperature.

Raises:

Type Description
TypeError

If energy or temperature is not a number, list, numpy array, or scipp Variable, or if energy_unit or temperature_unit is not a string or scipp Unit, or if divide_by_temperature is not a boolean.
ValueError

If temperature is negative, or if energy is a numpy array with more than 1 dimension, or if temperature is a scipp Variable that does not have a single dimension named 'temperature', or if energy is a scipp Variable that does not have a single dimension named 'energy'.
UnitError

If the provided energy_unit or temperature_unit is invalid, or if the units of energy or temperature cannot be converted to the expected units.
ZeroDivisionError

If divide_by_temperature is True and temperature is zero.

Examples:

detailed_balance_factor(1.0, 300) # 1 meV at 300 K detailed_balance_factor( ... energy=[1.0, 2.0], ... temperature=300, ... energy_unit='microeV', ... temperature_unit='K', ... divide_by_temperature=False, ... )

Source code in src/easydynamics/utils/detailed_balance.py 
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def detailed_balance_factor(
    energy: int | float | list | np.ndarray | sc.Variable,
    temperature: int | float | sc.Variable | Parameter,
    energy_unit: str | sc.Unit = 'meV',
    temperature_unit: str | sc.Unit = 'K',
    divide_by_temperature: bool = True,
) -> np.ndarray:
    r"""
    Compute the detailed balance factor (DBF):
    $$
    DBF(E, T) = E(n(E)+1)=\frac{E}{(1 - e^{-E / (k_B*T)})}},
    $$
    where $n(E)$ is the Bose-Einstein distribution, $E$ is the energy
    transfer, and $T$ is the temperature. $k_B$ is the Boltzmann
    constant.
    If divide_by_temperature is True, the result is normalized by
    $k_B*T$ to have value 1 at $E=0$.

    Args:
        energy (int | float | list | np.ndarray | sc.Variable): The energy
            transfer. If number, assumed to be in meV unless energy_unit
            is set.
        temperature (int | float | sc.Variable | Parameter): The
            temperature. If number, assumed to be in K unless
            temperature_unit is set.
        energy_unit (str | sc.Unit, default='meV'): Unit for energy if energy is
            given as a number or list. Default is 'meV'
        temperature_unit (str | sc.Unit, default='K'): Unit for temperature if
            temperature is given as a number. Default is 'K'
        divide_by_temperature (bool, default=True): If True, divide the result
            by $k_B*T$ to make it dimensionless and have value 1 at E=0.
            Default is True.

    Returns:
        np.ndarray:  Detailed balance factor evaluated at the
            given energy and temperature.

    Raises:
        TypeError: If energy or temperature is not a number, list,
            numpy array, or scipp Variable, or if energy_unit or
            temperature_unit is not a string or scipp Unit,
            or if divide_by_temperature is not a boolean.
        ValueError: If temperature is negative, or if energy is a numpy
            array with more than 1 dimension, or if temperature is a
            scipp Variable that does not have a single dimension named
            'temperature', or if energy is a scipp Variable that does
            not have a single dimension named 'energy'.
        UnitError: If the provided energy_unit or temperature_unit is
            invalid, or if the units of energy or temperature cannot be
            converted to the expected units.
        ZeroDivisionError: If divide_by_temperature is True and temperature is zero.

    Examples:
    >>> detailed_balance_factor(1.0, 300)  # 1 meV at 300 K
    >>> detailed_balance_factor(
    ...     energy=[1.0, 2.0],
    ...     temperature=300,
    ...     energy_unit='microeV',
    ...     temperature_unit='K',
    ...     divide_by_temperature=False,
    ... )
    """

    # Input validation
    if not isinstance(divide_by_temperature, bool):
        raise TypeError('divide_by_temperature must be True or False.')

    if not isinstance(energy_unit, (str, sc.Unit)):
        raise TypeError('energy_unit must be a string or scipp.Unit.')

    if not isinstance(temperature_unit, (str, sc.Unit)):
        raise TypeError('temperature_unit must be a string or scipp.Unit.')

    # Convert temperature and energy to sc variables
    # to make units easy to handle
    temperature = _convert_to_scipp_variable(
        value=temperature, unit=temperature_unit, name='temperature'
    )

    if temperature.value < 0:
        raise ValueError('Temperature must be non-negative.')

    energy = _convert_to_scipp_variable(value=energy, unit=energy_unit, name='energy')

    # What if people give units that don't make sense?
    try:
        sc.to_unit(energy, unit='meV')
    except Exception as e:
        raise UnitError(
            f'The unit of energy is wrong: {energy.unit}: {e} Check that energy has a valid unit.'
        ) from e
    # We give users the option to specify the unit of the energy,
    # but if the input has a unit, they might clash
    if energy.unit != energy_unit:
        warnings.warn(
            f'Input energy has unit {energy.unit}, but energy_unit was set to {energy_unit}. '
            f'Using {energy.unit}.',
            stacklevel=2,
        )

    # Same for temperature
    try:
        sc.to_unit(temperature, unit='K')
    except Exception as e:
        raise UnitError(
            f'The unit of temperature is wrong: {temperature.unit}: {e} '
            f'Check that temperature has a valid unit.'
        ) from e

    if temperature.unit != temperature_unit:
        warnings.warn(
            f'Input temperature has unit {temperature.unit}, '
            f'but temperature_unit was set to {temperature_unit}. Using {temperature.unit}.',
            stacklevel=2,
        )

    # Zero temperature deserves special treatment.
    # Here, DBF is 0 for energy<0 and energy for energy>0
    if temperature.value == 0:
        if divide_by_temperature:
            raise ZeroDivisionError('Cannot divide by T when T = 0.')
        DBF = sc.where(energy < 0.0 * energy.unit, 0.0 * energy.unit, energy)

        return np.array([DBF.value]) if DBF.sizes == {} else DBF.values

    # Now work with finite temperatures.
    # Here, it helps to work with dimensionless x = energy/(kB*T),
    # where we have divided by kB*T
    # We first check if the units are OK.

    x = energy / (kB * temperature)

    x = sc.to_unit(x, unit='1')  # Make sure the unit is 1 and not e.g. 1e3

    # Now compute DBF. First handle small and large x, then general.

    # Small x (small energy and/or high temperature): Taylor expansion.
    # Works and is needed for both positive and negative energies
    small = sc.abs(x) < SMALL_THRESHOLD

    DBF = sc.where(small, 1 + x / 2 + x**2 / 12, sc.zeros_like(x))

    # Large x (large positive energy and/or low temperature):
    # asymptotic form. Only needed for positive energies.
    large = x > LARGE_THRESHOLD
    DBF = sc.where(large, x, DBF)

    # General case: exact formula
    mid = sc.logical_not(small) & sc.logical_not(large)
    DBF = sc.where(mid, x / (1 - sc.exp(-x)), DBF)  # zeros in x are handled by SMALL_THRESHOLD

    #
    if not divide_by_temperature:
        DBF = DBF * (kB * temperature)
        DBF = sc.to_unit(DBF, unit=energy.unit)

    return np.array([DBF.value]) if DBF.sizes == {} else DBF.values

detailed_balance_factor(energy, temperature, energy_unit='meV', temperature_unit='K', divide_by_temperature=True)

Compute the detailed balance factor (DBF): $$ DBF(E, T) = E(n(E)+1)=\frac{E}{(1 - e^{-E / (k_B*T)})}}, $$ where \(n(E)\) is the Bose-Einstein distribution, \(E\) is the energy transfer, and \(T\) is the temperature. \(k_B\) is the Boltzmann constant. If divide_by_temperature is True, the result is normalized by \(k_B*T\) to have value 1 at \(E=0\).

Parameters:

Name Type Description Default
energy int | float | list | ndarray | Variable

The energy transfer. If number, assumed to be in meV unless energy_unit is set.

 required
temperature int | float | Variable | Parameter

The temperature. If number, assumed to be in K unless temperature_unit is set.

 required
energy_unit str | sc.Unit, default='meV'

Unit for energy if energy is given as a number or list. Default is 'meV'

 'meV'
temperature_unit str | sc.Unit, default='K'

Unit for temperature if temperature is given as a number. Default is 'K'

 'K'
divide_by_temperature bool, default=True

If True, divide the result by \(k_B*T\) to make it dimensionless and have value 1 at E=0. Default is True.

 True

Returns:

Type Description
ndarray

np.ndarray: Detailed balance factor evaluated at the given energy and temperature.

Raises:

Type Description
TypeError

If energy or temperature is not a number, list, numpy array, or scipp Variable, or if energy_unit or temperature_unit is not a string or scipp Unit, or if divide_by_temperature is not a boolean.
ValueError

If temperature is negative, or if energy is a numpy array with more than 1 dimension, or if temperature is a scipp Variable that does not have a single dimension named 'temperature', or if energy is a scipp Variable that does not have a single dimension named 'energy'.
UnitError

If the provided energy_unit or temperature_unit is invalid, or if the units of energy or temperature cannot be converted to the expected units.
ZeroDivisionError

If divide_by_temperature is True and temperature is zero.

Examples:

detailed_balance_factor(1.0, 300) # 1 meV at 300 K detailed_balance_factor( ... energy=[1.0, 2.0], ... temperature=300, ... energy_unit='microeV', ... temperature_unit='K', ... divide_by_temperature=False, ... )

Source code in src/easydynamics/utils/detailed_balance.py 
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def detailed_balance_factor(
    energy: int | float | list | np.ndarray | sc.Variable,
    temperature: int | float | sc.Variable | Parameter,
    energy_unit: str | sc.Unit = 'meV',
    temperature_unit: str | sc.Unit = 'K',
    divide_by_temperature: bool = True,
) -> np.ndarray:
    r"""
    Compute the detailed balance factor (DBF):
    $$
    DBF(E, T) = E(n(E)+1)=\frac{E}{(1 - e^{-E / (k_B*T)})}},
    $$
    where $n(E)$ is the Bose-Einstein distribution, $E$ is the energy
    transfer, and $T$ is the temperature. $k_B$ is the Boltzmann
    constant.
    If divide_by_temperature is True, the result is normalized by
    $k_B*T$ to have value 1 at $E=0$.

    Args:
        energy (int | float | list | np.ndarray | sc.Variable): The energy
            transfer. If number, assumed to be in meV unless energy_unit
            is set.
        temperature (int | float | sc.Variable | Parameter): The
            temperature. If number, assumed to be in K unless
            temperature_unit is set.
        energy_unit (str | sc.Unit, default='meV'): Unit for energy if energy is
            given as a number or list. Default is 'meV'
        temperature_unit (str | sc.Unit, default='K'): Unit for temperature if
            temperature is given as a number. Default is 'K'
        divide_by_temperature (bool, default=True): If True, divide the result
            by $k_B*T$ to make it dimensionless and have value 1 at E=0.
            Default is True.

    Returns:
        np.ndarray:  Detailed balance factor evaluated at the
            given energy and temperature.

    Raises:
        TypeError: If energy or temperature is not a number, list,
            numpy array, or scipp Variable, or if energy_unit or
            temperature_unit is not a string or scipp Unit,
            or if divide_by_temperature is not a boolean.
        ValueError: If temperature is negative, or if energy is a numpy
            array with more than 1 dimension, or if temperature is a
            scipp Variable that does not have a single dimension named
            'temperature', or if energy is a scipp Variable that does
            not have a single dimension named 'energy'.
        UnitError: If the provided energy_unit or temperature_unit is
            invalid, or if the units of energy or temperature cannot be
            converted to the expected units.
        ZeroDivisionError: If divide_by_temperature is True and temperature is zero.

    Examples:
    >>> detailed_balance_factor(1.0, 300)  # 1 meV at 300 K
    >>> detailed_balance_factor(
    ...     energy=[1.0, 2.0],
    ...     temperature=300,
    ...     energy_unit='microeV',
    ...     temperature_unit='K',
    ...     divide_by_temperature=False,
    ... )
    """

    # Input validation
    if not isinstance(divide_by_temperature, bool):
        raise TypeError('divide_by_temperature must be True or False.')

    if not isinstance(energy_unit, (str, sc.Unit)):
        raise TypeError('energy_unit must be a string or scipp.Unit.')

    if not isinstance(temperature_unit, (str, sc.Unit)):
        raise TypeError('temperature_unit must be a string or scipp.Unit.')

    # Convert temperature and energy to sc variables
    # to make units easy to handle
    temperature = _convert_to_scipp_variable(
        value=temperature, unit=temperature_unit, name='temperature'
    )

    if temperature.value < 0:
        raise ValueError('Temperature must be non-negative.')

    energy = _convert_to_scipp_variable(value=energy, unit=energy_unit, name='energy')

    # What if people give units that don't make sense?
    try:
        sc.to_unit(energy, unit='meV')
    except Exception as e:
        raise UnitError(
            f'The unit of energy is wrong: {energy.unit}: {e} Check that energy has a valid unit.'
        ) from e
    # We give users the option to specify the unit of the energy,
    # but if the input has a unit, they might clash
    if energy.unit != energy_unit:
        warnings.warn(
            f'Input energy has unit {energy.unit}, but energy_unit was set to {energy_unit}. '
            f'Using {energy.unit}.',
            stacklevel=2,
        )

    # Same for temperature
    try:
        sc.to_unit(temperature, unit='K')
    except Exception as e:
        raise UnitError(
            f'The unit of temperature is wrong: {temperature.unit}: {e} '
            f'Check that temperature has a valid unit.'
        ) from e

    if temperature.unit != temperature_unit:
        warnings.warn(
            f'Input temperature has unit {temperature.unit}, '
            f'but temperature_unit was set to {temperature_unit}. Using {temperature.unit}.',
            stacklevel=2,
        )

    # Zero temperature deserves special treatment.
    # Here, DBF is 0 for energy<0 and energy for energy>0
    if temperature.value == 0:
        if divide_by_temperature:
            raise ZeroDivisionError('Cannot divide by T when T = 0.')
        DBF = sc.where(energy < 0.0 * energy.unit, 0.0 * energy.unit, energy)

        return np.array([DBF.value]) if DBF.sizes == {} else DBF.values

    # Now work with finite temperatures.
    # Here, it helps to work with dimensionless x = energy/(kB*T),
    # where we have divided by kB*T
    # We first check if the units are OK.

    x = energy / (kB * temperature)

    x = sc.to_unit(x, unit='1')  # Make sure the unit is 1 and not e.g. 1e3

    # Now compute DBF. First handle small and large x, then general.

    # Small x (small energy and/or high temperature): Taylor expansion.
    # Works and is needed for both positive and negative energies
    small = sc.abs(x) < SMALL_THRESHOLD

    DBF = sc.where(small, 1 + x / 2 + x**2 / 12, sc.zeros_like(x))

    # Large x (large positive energy and/or low temperature):
    # asymptotic form. Only needed for positive energies.
    large = x > LARGE_THRESHOLD
    DBF = sc.where(large, x, DBF)

    # General case: exact formula
    mid = sc.logical_not(small) & sc.logical_not(large)
    DBF = sc.where(mid, x / (1 - sc.exp(-x)), DBF)  # zeros in x are handled by SMALL_THRESHOLD

    #
    if not divide_by_temperature:
        DBF = DBF * (kB * temperature)
        DBF = sc.to_unit(DBF, unit=energy.unit)

    return np.array([DBF.value]) if DBF.sizes == {} else DBF.values

utils